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<h2>HL Paper 3</h2><div class="specification">
<p>An electron and a positron have identical speeds but are travelling in opposite directions. Their collision results in the annihilation of both particles and the production of two photons of identical energy. The initial kinetic energy of the electron is 2.00 MeV.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain, in terms of a conservation law, why two photons need to be created.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the speed of the incoming electron.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the energy and the momentum for each photon after the collision.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>as the total initial momentum is zero, it must be zero after the collision</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>2 = (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\gamma ">
<mi>γ</mi>
</math></span>–1)m<sub>0</sub>c<sup>2</sup> = (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\gamma ">
<mi>γ</mi>
</math></span>–1) 0.511</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\gamma ">
<mi>γ</mi>
</math></span> = 4.91</p>
<p><em>v</em> = 0.978<em>c</em></p>
<p> </p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>«2 + 2 + 2 × 0.511 = 5.02 MeV» so each photon is 2.51«MeV»<br><br><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="p = \frac{E}{c} = 2.51">
<mi>p</mi>
<mo>=</mo>
<mfrac>
<mi>E</mi>
<mi>c</mi>
</mfrac>
<mo>=</mo>
<mn>2.51</mn>
</math></span> «MeVc<sup>–1</sup>»</p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>The global positioning system (GPS) uses satellites that orbit the Earth. The satellites transmit information to Earth using accurately known time signals derived from atomic clocks on the satellites. The time signals need to be corrected due to the gravitational redshift that occurs because the satellites are at a height of 20 Mm above the surface of the Earth.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The gravitational field strength at 20 Mm above the surface of the Earth is about 0.6 N kg<sup>–1</sup>. Estimate the time correction per day needed to the time signals, due to the gravitational redshift.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Suggest, whether your answer to (a) underestimates <strong>or</strong> overestimates the correction required to the time signal.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\Delta f}}{f} = \frac{{gh}}{{{c^2}}}">
<mfrac>
<mrow>
<mi mathvariant="normal">Δ</mi>
<mi>f</mi>
</mrow>
<mi>f</mi>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mi>g</mi>
<mi>h</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> so <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\Delta f = \frac{{0.6 \times 20000000}}{{{{\left( {3 \times {{10}^8}} \right)}^2}}} = 1.3 \times {10^{ - 10}}">
<mi mathvariant="normal">Δ</mi>
<mi>f</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>0.6</mn>
<mo>×</mo>
<mn>20000000</mn>
</mrow>
<mrow>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>3</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mn>8</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1.3</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mrow>
<mo>−</mo>
<mn>10</mn>
</mrow>
</msup>
</mrow>
</math></span></p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\Delta f}}{f} = \frac{{\Delta t}}{t}">
<mfrac>
<mrow>
<mi mathvariant="normal">Δ</mi>
<mi>f</mi>
</mrow>
<mi>f</mi>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mi mathvariant="normal">Δ</mi>
<mi>t</mi>
</mrow>
<mi>t</mi>
</mfrac>
</math></span></p>
<p>1.3 × 10<sup>–10 </sup>× 24 × 3600 = 1.15×10<sup>–5</sup> «s» «running fast»</p>
<p><em>Award <strong>[3 max]</strong> if for g 0.6 <strong>OR</strong> 9.8 <strong>OR</strong> average of 0.6 and 9.8 is used.</em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>ALTERNATIVE 1 </strong></em></p>
<p><em>g</em> is not constant through ∆<em>h</em> so value determined should be larger</p>
<p><em><em>Use ECF from (a)<br></em>Accept under or overestimate for SR argument.</em></p>
<p><em><strong>ALTERNATIVE 2 </strong></em></p>
<p>the satellite clock is affected by time dilation due to special relativity/its motion</p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State what is meant by the event horizon of a black hole.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the surface area <em>A</em> of the sphere corresponding to the event horizon is given by</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = \frac{{16\pi {G^2}{M^2}}}{{{c^4}}}">
<mi>A</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>16</mn>
<mi>π</mi>
<mrow>
<msup>
<mi>G</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mi>M</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mi>c</mi>
<mn>4</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Suggest why the surface area of the event horizon can never decrease.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The diagram shows a box that is falling freely in the gravitational field of a planet.</p>
<p><img src="data:image/png;base64,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"></p>
<p>A photon of frequency <em>f</em> is emitted from the floor of the box and is received at the ceiling. State and explain the frequency of the photon measured at the ceiling.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>the surface at which the escape speed is the speed for light<br><em><strong>OR</strong></em><br>the surface from which nothing/not even light can escape to the outside<br><em><strong>OR</strong></em><br>the surface of a sphere whose radius is the Schwarzschild radius</p>
<p> </p>
<p><em>Accept distance as alternative to surface.</em></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>use of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = 4\pi {R^2}">
<mi>A</mi>
<mo>=</mo>
<mn>4</mn>
<mi>π</mi>
<mrow>
<msup>
<mi>R</mi>
<mn>2</mn>
</msup>
</mrow>
</math></span> and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="R = \frac{{2GM}}{{{c^2}}}">
<mi>R</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mi>G</mi>
<mi>M</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span></p>
<p>«to get <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = \frac{{16\pi {G^2}{M^2}}}{{{c^4}}}">
<mi>A</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>16</mn>
<mi>π</mi>
<mrow>
<msup>
<mi>G</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mi>M</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mi>c</mi>
<mn>4</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span>»</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>since mass and energy can never leave a black hole and <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="A = \frac{{16\pi {G^2}{M^2}}}{{{c^4}}}">
<mi>A</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>16</mn>
<mi>π</mi>
<mrow>
<msup>
<mi>G</mi>
<mn>2</mn>
</msup>
</mrow>
<mrow>
<msup>
<mi>M</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mrow>
<msup>
<mi>c</mi>
<mn>4</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span></p>
<p><em><strong>OR</strong></em></p>
<p>some statement that area is increasing with mass</p>
<p>«the area cannot decrease»</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>ALTERNATIVE 1 —</strong> (student/planet frame):</em></p>
<p>photon energy/frequency decreases with height<br><em><strong>OR</strong></em><br>there is a gravitational redshift</p>
<p>detector in ceiling is approaching photons so Doppler blue shift</p>
<p>two effects cancel/frequency unchanged</p>
<p><em><strong>ALTERNATIVE 2</strong> – (box frame):</em></p>
<p>by equivalence principle box is an inertial frame</p>
<p>so no force on photons</p>
<p>so no redshift/frequency unchanged</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.iii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>In an experiment a source of iron-57 emits gamma rays of energy 14.4 ke V. A detector placed 22.6 m vertically above the source measures the frequency of the gamma rays.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the expected shift in frequency between the emitted and the detected gamma rays.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain whether the detected frequency would be greater or less than the emitted frequency.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><em>f</em> = «<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{E}{h} = ">
<mfrac>
<mi>E</mi>
<mi>h</mi>
</mfrac>
<mo>=</mo>
</math></span>» <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{14\,400 \times 1.6 \times {{10}^{ - 19}}}}{{6.63 \times {{10}^{ - 34}}}}">
<mfrac>
<mrow>
<mn>14</mn>
<mspace width="thinmathspace"></mspace>
<mn>400</mn>
<mo>×</mo>
<mn>1.6</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>19</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mn>6.63</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>34</mn>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> = «3.475 × 10<sup>18 </sup>Hz»</p>
<p>Δ<em>f = </em>«<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{g \times \Delta h \times f}}{{{c^2}}} \approx ">
<mfrac>
<mrow>
<mi>g</mi>
<mo>×</mo>
<mi mathvariant="normal">Δ</mi>
<mi>h</mi>
<mo>×</mo>
<mi>f</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>≈</mo>
</math></span>» 8550 «Hz»</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>«as the photon moves away from the Earth, » it has to spend energy to overcome the gravitational field</p>
<p>since <em>E = h</em><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,">
<mspace width="thinmathspace"></mspace>
</math></span><em>f</em>, the detected frequency would be lower «than the emitted frequency»</p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A deuterium <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mmultiscripts><mi mathvariant="normal">H</mi><mprescripts></mprescripts><mn>1</mn><mn>2</mn></mmultiscripts></mfenced></math> nucleus (rest mass <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>014</mn><mo> </mo><mi mathvariant="normal">u</mi></math>) is accelerated by a potential difference of <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>700</mn><mo>×</mo><msup><mn>10</mn><mn>2</mn></msup><mo> </mo><mi>MV</mi></math>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Define rest mass.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the total energy of the deuterium particle in <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>MeV</mi></math>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>In relativistic reactions the mass of the products may be less than the mass of the reactants. Suggest what happens to the missing mass.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="fontstyle0">invariant mass<br></span><span class="fontstyle2"><em><strong>OR</strong></em><br></span><span class="fontstyle0">mass of object when not in motion/in object’s rest frame </span><span class="fontstyle3">✓</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="fontstyle0">«</span><span class="fontstyle1">rest energy =</span><span class="fontstyle0">» <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>.</mo><mn>014</mn><mo>×</mo><mn>931</mn><mo>.</mo><mn>5</mn></mrow></mfenced><mo> </mo><mo>«</mo><mi>MeV</mi><mo>»</mo></math></span><span class="fontstyle0"> </span><span class="fontstyle3">✓<br></span></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>«</mo><msub><mi>E</mi><mi mathvariant="normal">T</mi></msub><mo>=</mo><mi>KE</mi><mo>+</mo><mi>rest</mi><mo> </mo><mi>energy</mi><mo>=</mo><mn>270</mn><mo>.</mo><mn>0</mn><mo>+</mo><mfenced><mrow><mn>2</mn><mo>.</mo><mn>014</mn><mo>×</mo><mn>931</mn><mo>.</mo><mn>5</mn></mrow></mfenced><mo>=</mo><mo>»</mo><mo> </mo><mn>2146</mn><mo> </mo><mo>«</mo><mi>MeV</mi><mo>»</mo></math><span class="fontstyle0"> </span><span class="fontstyle3">✓</span></p>
<p> </p>
<p><em><span class="fontstyle4">Final answer accept <math xmlns="http://www.w3.org/1998/Math/MathML"><mn mathvariant="italic">3</mn><mo mathvariant="italic">.</mo><mn mathvariant="italic">443</mn><mo mathvariant="italic">×</mo><msup><mn mathvariant="italic">10</mn><mrow><mo mathvariant="italic">-</mo><mn mathvariant="italic">10</mn></mrow></msup><mi>J</mi></math></span><span class="fontstyle4"> if unit given </span></em></p>
<p><em><span class="fontstyle4">Award </span><strong><span class="fontstyle5">[2] </span></strong><span class="fontstyle4">marks for a bald correct answer.</span></em></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="fontstyle0">is converted to energy </span><span class="fontstyle2">✓</span></p>
<p><span class="fontstyle0">as kinetic energy of the products </span><span class="fontstyle2">✓</span></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>The definition of rest mass proved to be known by most candidates.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Most were able to score full marks. A few candidates messed up by trying to convert to J, although some were successful with this additional difficulty.</p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Answers scored at least one or usually both marks.</p>
<div class="question_part_label">c.</div>
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<br><hr><br><div class="specification">
<p>An observer A is on the surface of planet X. Observer B is in a stationary spaceship above the surface of planet X.</p>
<p>Observer A sends a beam of light with a frequency 500 THz towards observer B. When observer B receives the light he observes that the frequency has changed by Δ<em>f</em>.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-08-13_om_15.27.24.png" alt="M18/4/PHYSI/HP3/ENG/TZ1/07_01"></p>
<p>Observer B then sends a signal with frequency 1500 THz towards observer A.</p>
<p style="text-align: center;"><img src="images/Schermafbeelding_2018-08-13_om_15.28.04.png" alt="M18/4/PHYSI/HP3/ENG/TZ1/07_02"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the shift in frequency observed by A in terms of Δ<em>f</em>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the gravitational field strength on the surface of planet X.</p>
<p> The following data is given:</p>
<p> Δ<em>f </em>= 170 Hz.</p>
<p>The distance between observer A and B is 10 km. </p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Observer A now sends a beam of light initially parallel to the surface of the planet.</p>
<p><img src="images/Schermafbeelding_2018-08-13_om_16.50.21.png" alt="M18/4/PHYSI/HP3/ENG/TZ1/07.c"></p>
<p>Explain why the path of the light is curved.</p>
<div class="marks">[2]</div>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>Δ<em>f</em> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext=" \propto ">
<mo>∝</mo>
</math></span> <em>f</em></p>
<p>therefore the change is <strong>«</strong>–<strong>»</strong>3Δ<em>f</em></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>g</em> = <strong>«</strong><em>c</em><sup>2</sup> <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{\Delta f}}{{f\Delta h}}">
<mfrac>
<mrow>
<mi mathvariant="normal">Δ</mi>
<mi>f</mi>
</mrow>
<mrow>
<mi>f</mi>
<mi mathvariant="normal">Δ</mi>
<mi>h</mi>
</mrow>
</mfrac>
</math></span> =<strong>»</strong> (3 × 10<sup>8</sup>)<sup>2</sup><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{170}}{{5.0 \times {{10}^{14}} \times 10\,000}}">
<mfrac>
<mrow>
<mn>170</mn>
</mrow>
<mrow>
<mn>5.0</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mn>14</mn>
</mrow>
</msup>
</mrow>
<mo>×</mo>
<mn>10</mn>
<mspace width="thinmathspace"></mspace>
<mn>000</mn>
</mrow>
</mfrac>
</math></span></p>
<p><em>g</em> = 3.1 <strong>«</strong>ms<sup>–2</sup><strong>»</strong></p>
<p> </p>
<p><em>If POT mistake, award </em><strong><em>[0]</em></strong><em>.</em></p>
<p><em>Award </em><strong><em>[2] </em></strong><em>for BCA.</em></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>the mass of the planet warps spacetime around itself</p>
<p>the light will follow the shortest path in spacetime <strong>«</strong>which is curved<strong>»</strong></p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">c.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">c.</div>
</div>
<br><hr><br><div class="specification">
<p>In the Pound–Rebka–Snider experiment, a source of gamma rays was placed <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>22</mn><mo>.</mo><mn>6</mn><mo> </mo><mi mathvariant="normal">m</mi></math> vertically above a gamma ray detector, in a tower on Earth.</p>
<p style="text-align: center;"><img src="data:image/png;base64,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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the fractional change in frequency of the gamma rays at the detector.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain the cause of the frequency shift for the gamma rays in your answer in (a) in the Earth’s gravitational field.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b(i).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain the cause of the frequency shift for the gamma rays in your answer in (a) if the tower and detector were accelerating <strong>towards</strong> the gamma rays in free space.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b(ii).</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>«</mo><mfrac><mrow><mo>∆</mo><mi>f</mi></mrow><mi>f</mi></mfrac><mo>=</mo><mfrac><mrow><mi>g</mi><mo>∆</mo><mi>h</mi></mrow><msup><mi>c</mi><mn>2</mn></msup></mfrac><mo>=</mo><mfrac><mrow><mn>9</mn><mo>.</mo><mn>81</mn><mo>×</mo><mn>22</mn><mo>.</mo><mn>6</mn></mrow><msup><mi>c</mi><mn>2</mn></msup></mfrac><mo>»</mo><mfrac><mrow><mo>∆</mo><mi>f</mi></mrow><mi>f</mi></mfrac><mo>=</mo><mn>2</mn><mo>.</mo><mn>46</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>15</mn></mrow></msup></math> ✓</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="fontstyle0"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>G</mi><mi>P</mi><mi>E</mi></math> </span><span class="fontstyle2">gained by photons so </span><span class="fontstyle0"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi></math> </span><span class="fontstyle2">increases </span><span class="fontstyle3">✓</span></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mi>h</mi><mi>f</mi></math><span class="fontstyle2">, so frequency increases </span><span class="fontstyle3">✓</span></p>
<div class="question_part_label">b(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="fontstyle0">gamma rays travel at </span><span class="fontstyle2"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>c</mi></math> </span><span class="fontstyle3">✓</span></p>
<p><span class="fontstyle0">detector accelerates towards source so </span><span class="fontstyle4">«</span><span class="fontstyle0">by Doppler effect</span><span class="fontstyle4">» </span><span class="fontstyle5"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>λ</mi></math> </span><span class="fontstyle0">reduced so frequency increases </span><span class="fontstyle3">✓</span></p>
<p> </p>
<p><em><span class="fontstyle0">Award </span><strong><span class="fontstyle2">[1 max] </span></strong><span class="fontstyle0">for reference to principle of equivalence without further explanation.</span></em></p>
<div class="question_part_label">b(ii).</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>The calculation was easily done by most candidates.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>A simple conceptual explanation in terms of the energy changes was not present.</p>
<div class="question_part_label">b(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Some candidates did score marks although it was not as simple as anticipated for a classical question.</p>
<div class="question_part_label">b(ii).</div>
</div>
<br><hr><br><div class="specification">
<p>Two protons, travelling in opposite directions, collide. Each has a total energy of 3.35 GeV.</p>
</div>
<div class="specification">
<p>As a result of the collision, the protons are annihilated and three particles, a proton, a neutron, and a pion are created. The pion has a rest mass of 140 MeV c<sup>–2</sup>. The total energy of the emitted proton and neutron from the interaction is 6.20 GeV.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the gamma (<em>γ</em>) factor for one of the protons.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine, in terms of MeV c<sup>–1</sup>, the momentum of the pion.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The diagram shows the paths of the incident protons together with the proton and neutron created in the interaction. On the diagram, draw the path of the pion.</p>
<p><img src="data:image/png;base64,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"></p>
<div class="marks">[1]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><em>γ </em><strong>«</strong>= <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{3350}}{{938}}">
<mfrac>
<mrow>
<mn>3350</mn>
</mrow>
<mrow>
<mn>938</mn>
</mrow>
</mfrac>
</math></span><strong>»</strong> = 3.37</p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>energy of pion = (3350 × 2) – 6200 = 500 <strong>«</strong>MeV<strong>»</strong></p>
<p>500<sup>2</sup> = <em>p</em><sup>2</sup><em>c</em><sup>2</sup> + 140<sup>2</sup></p>
<p><em>p =</em> 480 <strong>«</strong>MeV c<sup>–1</sup><strong>»</strong></p>
<p> </p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>path of pion constructed in direction around 4–5 o’clock by eye</p>
<p> </p>
<p><img src="images/Schermafbeelding_2018-08-14_om_09.50.44.png" alt="M18/4/PHYSI/HP3/ENG/TZ2/06.b.ii/M"></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A positive pion decays into a positive muon and a neutrino.</p>
<p><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="{\pi ^ + } \to {\mu ^ + } + {v_\mu }">
<mrow>
<msup>
<mi>π<!-- π --></mi>
<mo>+</mo>
</msup>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mrow>
<msup>
<mi>μ<!-- μ --></mi>
<mo>+</mo>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msub>
<mi>v</mi>
<mi>μ<!-- μ --></mi>
</msub>
</mrow>
</math></span></p>
<p>The momentum of the muon is measured to be 29.8 MeV c<sup>–1</sup> in a laboratory reference frame in which the pion is at rest. The rest mass of the muon is 105.7 MeV c<sup>–2</sup> and the mass of the neutrino can be assumed to be zero.</p>
</div>
<div class="specification">
<p>For the laboratory reference frame</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>write down the momentum of the neutrino.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>show that the energy of the pion is about 140 MeV.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the rest mass of the pion with an appropriate unit.</p>
<div class="marks">[1]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><strong>«</strong>–<strong>»</strong>29.8 <strong>«</strong>MeVc<sup>–1</sup><strong>»</strong></p>
<p><em><strong>[1 mark]</strong></em></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>E<sub>π</sub></em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {p_\mu ^2{c^2} + m_\mu ^2{c^4}} ">
<msqrt>
<msubsup>
<mi>p</mi>
<mi>μ</mi>
<mn>2</mn>
</msubsup>
<mrow>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<msubsup>
<mi>m</mi>
<mi>μ</mi>
<mn>2</mn>
</msubsup>
<mrow>
<msup>
<mi>c</mi>
<mn>4</mn>
</msup>
</mrow>
</msqrt>
</math></span> + <em>p<sub>v</sub>c</em> <strong><em>OR</em></strong> <em>E<sub>μ</sub></em> = 109.8 <strong>«</strong>MeV<strong>»</strong></p>
<p><em>E<sub>π</sub></em> = <strong>«</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {{{29.8}^2} + {{105.7}^2}} ">
<msqrt>
<mrow>
<msup>
<mrow>
<mn>29.8</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mn>105.7</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span> + 29.8 =<strong>»</strong> 139.6 <strong>«</strong>MeV<strong>»</strong></p>
<p> </p>
<p><em>Final value to at least 3 sig figs required for mark.</em></p>
<p><strong><em>[2 marks]</em></strong></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>139.6 MeVc<sup>–2</sup></p>
<p> </p>
<p><em>Units required.</em></p>
<p><em>Accept 140 MeVc<sup>–</sup></em><sup><em>2</em></sup><em>.</em></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>It is believed that a non-rotating supermassive black hole is likely to exist near the centre of our galaxy. This black hole has a mass equivalent to 3.6 million times that of the Sun.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Outline what is meant by the event horizon of a black hole.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the distance of the event horizon of the black hole from its centre.</p>
<p> Mass of Sun = 2 × 10<sup>30</sup> kg</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Star S-2 is in an elliptical orbit around a black hole. The distance of S-2 from the centre of the black hole varies between a few light-hours and several light-days. A periodic event on S-2 occurs every 5.0 s.</p>
<p><img src="images/Schermafbeelding_2018-08-14_om_10.00.36.png" alt="M18/4/PHYSI/HP3/ENG/TZ2/07.b"></p>
<p>Discuss how the time for the periodic event as measured by an observer on the Earth changes with the orbital position of S-2.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>boundary inside which events cannot be communicated to an outside observer</p>
<p><strong><em>OR</em></strong></p>
<p>distance/surface at which escape velocity = <em>c</em></p>
<p> </p>
<p><em>OWTTE</em></p>
<p><strong><em>[1 mark]</em></strong></p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>mass of black hole = 7.2 × 10<sup>36</sup> <strong>«</strong>kg<strong>»</strong></p>
<p><strong>«</strong><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2GM}}{{{c^2}}}">
<mfrac>
<mrow>
<mn>2</mn>
<mi>G</mi>
<mi>M</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</math></span> =<strong>»</strong> 1 × 10<sup>10</sup> <strong>«</strong>m<strong>»</strong></p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>wherever S-2 is in orbit, time observed is longer than 5.0 s</p>
<p>when closest to the star S-2 periodic time dilated more than when at greatest distance</p>
<p>Justification using formula or time is more dilated in stronger gravitational fields</p>
<p> </p>
<p><em><strong>[2 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="background-color: #ffffff;">A proton has a total energy 1050 MeV after being accelerated from rest through a potential difference <em>V</em>.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="background-color:#ffffff;">Define<em> total energy</em>.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="background-color:#ffffff;">Determine the momentum of the proton.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">bi.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="background-color:#ffffff;">Determine the speed of the proton.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">bii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="background-color:#ffffff;">Calculate the potential difference <em>V.</em></span></p>
<div class="marks">[1]</div>
<div class="question_part_label">biii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><span style="background-color:#ffffff;">total energy is the sum of the rest energy and the kinetic energy</span><em><span style="background-color:#ffffff;"> ✔</span></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;">«<em>p</em><sup><span style="font-size:small;">2</span></sup> <em>c</em><sup><span style="font-size:small;">2</span></sup> = 1050<sup><span style="font-size:small;">2</span></sup>–938<sup><span style="font-size:small;">2</span></sup> therefore» p=472«MeVc<sup><span style="font-size:small;">–</span></sup><sup><span style="font-size:small;">1</span></sup>»<em><span style="background-color:#ffffff;">✔</span></em></p>
<div class="question_part_label">bi.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><span style="background-color:#ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\gamma = \frac{{1050}}{{938}} = 1.12">
<mi>γ</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>1050</mn>
</mrow>
<mrow>
<mn>938</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>1.12</mn>
</math></span> ✔</span></p>
<p style="text-align:left;"><span style="background-color:#ffffff;"><span style="background-color:#ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v = 0.45c">
<mi>v</mi>
<mo>=</mo>
<mn>0.45</mn>
<mi>c</mi>
</math></span></span></span></p>
<p style="text-align:left;"><em><strong>OR</strong></em></p>
<p style="text-align:left;"><span style="background-color:#ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="V = 1.35 \times {10^8}">
<mi>V</mi>
<mo>=</mo>
<mn>1.35</mn>
<mo>×</mo>
<mrow>
<msup>
<mn>10</mn>
<mn>8</mn>
</msup>
</mrow>
</math></span>«ms<sup>–1</sup>» <span style="display:inline !important;float:none;background-color:#ffffff;color:#000000;font-family:Verdana , Arial , Helvetica , sans-serif;font-size:14px;font-style:normal;font-variant:normal;font-weight:400;letter-spacing:normal;text-align:left;text-decoration:none;text-indent:0px;white-space:normal;">✔</span></span></p>
<p style="text-align:left;"> </p>
<div class="question_part_label">bii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><span style="background-color:#ffffff;"><em>V</em> = 112 «MV» ✔</span></p>
<div class="question_part_label">biii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Generally well answered by most candidates. A common mistake was to define the total energy in the context of classical mechanics.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Most candidates seemed to have the right starting points but mistakes were often made in attempting to convert units. The energy-momentum equation is generally best answered using only ‘MeV’ based units.</p>
<div class="question_part_label">bi.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>An easy calculation, that was generally well answered.</p>
<div class="question_part_label">bii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Very few candidates realised that this question required a simple calculation using eV= KE = (γ-1)E.</p>
<div class="question_part_label">biii.</div>
</div>
<br><hr><br><div class="specification">
<p>The particle omega minus (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\Omega ^ - }">
<mrow>
<msup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>−<!-- − --></mo>
</msup>
</mrow>
</math></span>) decays at rest into a neutral pion (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\pi ^0}">
<mrow>
<msup>
<mi>π<!-- π --></mi>
<mn>0</mn>
</msup>
</mrow>
</math></span>) and the xi baryon (<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\Xi ^ - }">
<mrow>
<msup>
<mi mathvariant="normal">Ξ<!-- Ξ --></mi>
<mo>−<!-- − --></mo>
</msup>
</mrow>
</math></span>) according to</p>
<p style="text-align: center;"><sup><span style="font-size: small; background-color: #ffffff;"><span class="mjpage mjpage__block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" alttext="{\Omega ^ - } \to {\pi ^0} + {\Xi ^ - }">
<mrow>
<msup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>−<!-- − --></mo>
</msup>
</mrow>
<mo stretchy="false">→<!-- → --></mo>
<mrow>
<msup>
<mi>π<!-- π --></mi>
<mn>0</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mi mathvariant="normal">Ξ<!-- Ξ --></mi>
<mo>−<!-- − --></mo>
</msup>
</mrow>
</math></span></span></sup></p>
<p>The pion momentum is 289.7 MeV c<sup>–1</sup>.</p>
<p>The rest masses of the particles are:</p>
<p style="text-align: center;"><span style="display: inline !important; float: none; background-color: #ffffff; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\Omega ^ - }">
<mrow>
<msup>
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>−<!-- − --></mo>
</msup>
</mrow>
</math></span></span>: 1672 MeV c<sup>–2</sup></p>
<p style="text-align: center;"><span style="display: inline !important; float: none; background-color: #ffffff; color: #000000; font-family: Verdana,Arial,Helvetica,sans-serif; font-size: 14px; font-style: normal; font-variant: normal; font-weight: 400; letter-spacing: normal; orphans: 2; text-align: left; text-decoration: none; text-indent: 0px; text-transform: none; -webkit-text-stroke-width: 0px; white-space: normal; word-spacing: 0px;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\pi ^0}">
<mrow>
<msup>
<mi>π<!-- π --></mi>
<mn>0</mn>
</msup>
</mrow>
</math></span></span>: 135.0 MeV c<sup>–2</sup></p>
<p style="text-align: center;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\Xi ^ - }">
<mrow>
<msup>
<mi mathvariant="normal">Ξ<!-- Ξ --></mi>
<mo>−<!-- − --></mo>
</msup>
</mrow>
</math></span>: 1321 MeV c<sup>–2</sup></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that energy is conserved in this decay.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the speed of the pion.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>momentum of xi baryon is also 289.7«MeVc<sup>−1</sup>» ✔</p>
<p>total energy of xi baryon and pion is <span style="background-color:#ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {{{289.7}^2} + {{1321}^2}} + \sqrt {{{289.7}^2} + {{135.0}^2}} = 1672">
<msqrt>
<mrow>
<msup>
<mrow>
<mn>289.7</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mn>1321</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mo>+</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mn>289.7</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mn>135.0</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mo>=</mo>
<mn>1672</mn>
</math></span></span> «MeV» ✔</p>
<p>which equals the rest energy of the omega ✔</p>
<p><em>Allow a backwards argument, assuming the energy is equal.</em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="background-color:#ffffff;"><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\gamma « = \frac{{\sqrt {{{289.7}^2} + {{135.0}^2}} }}{{135.0}} » = 2.367">
<mi>γ</mi>
<mrow>
<mo>«</mo>
</mrow>
<mo>=</mo>
<mfrac>
<mrow>
<msqrt>
<mrow>
<msup>
<mrow>
<mn>289.7</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mn>135.0</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</mrow>
<mrow>
<mn>135.0</mn>
</mrow>
</mfrac>
<mrow>
<mo>»</mo>
</mrow>
<mo>=</mo>
<mn>2.367</mn>
</math></span></span> ✔</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v «= \sqrt {1 - \frac{1}{{{{2.367}^2}}}} c » = 0.903c">
<mi>v</mi>
<mrow>
<mo>«</mo>
</mrow>
<mo>=</mo>
<msqrt>
<mn>1</mn>
<mo>−</mo>
<mfrac>
<mn>1</mn>
<mrow>
<mrow>
<msup>
<mrow>
<mn>2.367</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</msqrt>
<mi>c</mi>
<mrow>
<mo>»</mo>
</mrow>
<mo>=</mo>
<mn>0.903</mn>
<mi>c</mi>
</math></span> ✔</p>
<p><em>Award <strong>[2]</strong> for bald correct answer.</em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Energy conservation in a decay. This relativistic mechanics question was very challenging. Only a few candidates were able to calculate the total energy of the baryon and pion, despite being able to recognize the correct momentum of the baryon.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Energy conservation in a decay. This relativistic mechanics question was very challenging. Only a few candidates were able to calculate the total energy of the baryon and pion, despite being able to recognize the correct momentum of the baryon. The same for the speed of the pion in b), most of the attempts demonstrated by candidates were not relevant.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>An electron with total energy 1.50 MeV collides with a positron at rest. As a result two photons are produced. One photon moves in the same direction as the electron and the other in the opposite direction.</p>
</div>
<div class="specification">
<p>The momenta of the photons produced have magnitudes <em>p</em><sub>1</sub> and <em>p</em><sub>2</sub>. A student writes the following correct equations.</p>
<p style="padding-left: 210px;"><em>p</em><sub>1</sub> – <em>p</em><sub>2</sub> = 1.41 MeV c<sup>–1</sup></p>
<p style="padding-left: 210px;"><em>p</em><sub>1</sub> + <em>p</em><sub>2</sub> = 2.01 MeV c<sup>–1</sup></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Show that the momentum of the electron is 1.41 MeV c<sup>–1</sup>.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Explain the origin of each equation.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate, in MeV c<sup>–1</sup>, <em>p</em><sub>1</sub> and <em>p</em><sub>2</sub>.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="pc = \sqrt {{E^2} - {{\left( {m{c^2}} \right)}^2}} = \sqrt {{{150}^2} - {{0.511}^2}} ">
<mi>p</mi>
<mi>c</mi>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mi>E</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>m</mi>
<mrow>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mn>150</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mn>0.511</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
</math></span> «= 1.410 MeV» ✔</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>first equation is due to momentum conservation ✔</p>
<p>second equation is due to total energy conservation ✔</p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>adding 2<em>p</em><sub>1</sub> = 3.42 MeV c<sup>–1</sup> ⇒ <em>p</em><sub>1</sub> = 1.17 MeV c<sup>–1</sup> ✔</p>
<p><em>p</em><sub>2</sub> = 0.30 MeV c<sup>–1</sup> ✔</p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br><div class="specification">
<p>A probe launched from a spacecraft moves towards the event horizon of a black hole.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State what is meant by the event horizon of a black hole.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The mass of the black hole is 4.0 × 10<sup>36 </sup>kg. Calculate the Schwarzschild radius of the black hole.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The probe is stationary above the event horizon of the black hole in (a). The probe sends a radio pulse every 1.0 seconds (as measured by clocks on the probe). The spacecraft receives the pulses every 2.0 seconds (as measured by clocks on the spacecraft). Determine the distance of the probe from the centre of the black hole.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>the distance from the black hole at which the escape speed is the speed of light ✔</p>
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em>R</em><sub>s</sub> = «<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{2GM}}{{{c^2}}} = \frac{{2 \times 6.67 \times {{10}^{ - 11}} \times 4.0 \times {{10}^{36}}}}{{9.0 \times {{10}^{16}}}} = ">
<mfrac>
<mrow>
<mn>2</mn>
<mi>G</mi>
<mi>M</mi>
</mrow>
<mrow>
<mrow>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
<mfrac>
<mrow>
<mn>2</mn>
<mo>×</mo>
<mn>6.67</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mo>−</mo>
<mn>11</mn>
</mrow>
</msup>
</mrow>
<mo>×</mo>
<mn>4.0</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mn>36</mn>
</mrow>
</msup>
</mrow>
</mrow>
<mrow>
<mn>9.0</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mrow>
<mn>16</mn>
</mrow>
</msup>
</mrow>
</mrow>
</mfrac>
<mo>=</mo>
</math></span>» 5.9 × 10<sup>9</sup> «m» ✔</p>
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="2 = \frac{1}{{\sqrt {1 - \frac{{5.9 \times {{10}^9}}}{r}} }}">
<mn>2</mn>
<mo>=</mo>
<mfrac>
<mn>1</mn>
<mrow>
<msqrt>
<mn>1</mn>
<mo>−</mo>
<mfrac>
<mrow>
<mn>5.9</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mn>9</mn>
</msup>
</mrow>
</mrow>
<mi>r</mi>
</mfrac>
</msqrt>
</mrow>
</mfrac>
</math></span> ✔</p>
<p>rearranged to give <em>r</em></p>
<p><em><strong>OR</strong></em></p>
<p><em>r</em> = 1.33 × 5.9 × 10<sup>9</sup> «m» ✔</p>
<p><em>r</em> = 7.9 × 10<sup>9</sup> «m» ✔</p>
<p> </p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.ii.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A proton is accelerated from rest through a potential difference <em>V</em> to a speed of 0.86c.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the potential difference <em>V</em>.</p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>The proton collides with an antiproton moving with the same speed in the opposite direction. As a result both particles are annihilated and two photons of equal energy are produced.</p>
<p>Determine the momentum of one of the photons.</p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><em>γ</em> = 1.96</p>
<p><em>E</em><sub>k</sub> = (<em>γ</em> − 1) <em>m</em><sub>0</sub><em>c</em><sup>2</sup> = 900 «Me<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,">
<mspace width="thinmathspace"></mspace>
</math></span>V»</p>
<p><em>pd</em> ≈ 900 «MV»<br><br></p>
<p><em>Award <strong>[2 max]</strong> if Energy and Potential difference are not clearly distinguished, eg by the unit.</em></p>
<p><strong><em>[3 marks]</em></strong></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>energy of proton = <em>γ</em>mc<sup>2</sup> = 1838 «Me<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,">
<mspace width="thinmathspace"></mspace>
</math></span>V»</p>
<p>total energy available = energy of proton + energy of antiproton = 1838 + 1838 = 3676 «Me<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,">
<mspace width="thinmathspace"></mspace>
</math></span>V»</p>
<p>momentum of a one photon = Total energy / 2c = 1838 «Me<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,">
<mspace width="thinmathspace"></mspace>
</math></span>Vc<sup>–1</sup>»</p>
<p><em><strong>[3 marks]</strong></em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="background-color: #ffffff;">A rocket is accelerating upwards at 9.8 m s<sup>-2</sup> in deep space. A photon of energy 14.4 keV is emitted upwards from the bottom of the rocket and travels to a detector in the tip of the rocket 52.0 m above.</span></p>
<p><span style="background-color: #ffffff;"><img src="data:image/png;base64,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"></span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="background-color:#ffffff;">Explain why a change in frequency is expected for the photon detected at the top of the rocket.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="background-color:#ffffff;">Calculate the frequency change.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><span style="background-color:#ffffff;"><em><strong>ALTERNATIVE 1</strong></em><br></span></p>
<p style="text-align:left;"><span style="background-color:#ffffff;">detector accelerates/moves away from point of photon emission ✔<br></span></p>
<p style="text-align:left;"><span style="background-color:#ffffff;">so Doppler effect / redshift ✔<br></span></p>
<p style="text-align:left;"><span style="background-color:#ffffff;">so <em>f</em> decreases ✔<br></span></p>
<p style="text-align:left;"><span style="background-color:#ffffff;"><em><strong>ALTERNATIVE 2</strong></em><br></span></p>
<p style="text-align:left;"><span style="background-color:#ffffff;">equivalent to stationary rocket on earth’s surface ✔<br></span></p>
<p style="text-align:left;"><span style="background-color:#ffffff;">photons lose «gravitational» energy as they move upwards ✔<br></span></p>
<p style="text-align:left;"><span style="background-color:#ffffff;">h<em> f <strong>OR</strong> f</em> decreases ✔</span></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p style="text-align:left;"><img 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" width="460" height="153"></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Most candidates answered this question correctly using the equivalence principle and could show that the frequency would decrease.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Arithmetic mistakes were common at the different stages of the calculations even when the process used was correct.</p>
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="question">
<p>A lambda <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\Lambda ">
<mi mathvariant="normal">Λ</mi>
</math></span><sup>0</sup> particle at rest decays into a proton p and a pion <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\pi ^ - }">
<mrow>
<msup>
<mi>π</mi>
<mo>−</mo>
</msup>
</mrow>
</math></span> according to the reaction</p>
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\Lambda ">
<mi mathvariant="normal">Λ</mi>
</math></span><sup>0</sup> → p + <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π</mi>
</math></span><sup>–</sup></p>
<p>where the rest energy of p = 938 MeV and the rest energy of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\pi ">
<mi>π</mi>
</math></span><sup>–</sup> = 140 MeV.</p>
<p>The speed of the pion after the decay is 0.579<em>c</em>. For this speed <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\gamma ">
<mi>γ</mi>
</math></span> = 1.2265. Calculate the speed of the proton.</p>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question">
<p>pion momentum is <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\gamma mv = 1.2265 \times 140 \times 0.579 = 99.4">
<mi>γ</mi>
<mi>m</mi>
<mi>v</mi>
<mo>=</mo>
<mn>1.2265</mn>
<mo>×</mo>
<mn>140</mn>
<mo>×</mo>
<mn>0.579</mn>
<mo>=</mo>
<mn>99.4</mn>
</math></span> «MeV<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\,">
<mspace width="thinmathspace"></mspace>
</math></span>c<sup>–1</sup>»</p>
<p>use of momentum conservation to realize that produced particles have equal and opposite momenta</p>
<p>so for proton <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\gamma v = \frac{{99.4}}{{938}} = 0.106c">
<mi>γ</mi>
<mi>v</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mn>99.4</mn>
</mrow>
<mrow>
<mn>938</mn>
</mrow>
</mfrac>
<mo>=</mo>
<mn>0.106</mn>
<mi>c</mi>
</math></span></p>
<p>solving to get <em>v</em> = 0.105<em>c</em></p>
<p> </p>
<p><em>Accept pion momentum calculation using E <sup>2</sup> = p <sup>2</sup>c <sup>2</sup> +m <sup>2</sup>c <sup>4</sup>.<br></em></p>
<p><em>Award <strong>[2 max]</strong> for a non-relativistic answer of v = 0.0864c</em></p>
<p><strong><em>[4 marks]</em></strong></p>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question">
[N/A]
</div>
<br><hr><br><div class="specification">
<p>The <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\Lambda ^0}">
<mrow>
<msup>
<mi mathvariant="normal">Λ<!-- Λ --></mi>
<mn>0</mn>
</msup>
</mrow>
</math></span> (Lambda) particle decays spontaneously into a proton and a negatively charged pion of rest mass 140 MeV c<sup>–2</sup>. After the decay, the particles are moving in the same direction with a proton momentum of 630 MeV c<sup>–1</sup> and a pion momentum of 270 MeV c<sup>–1</sup>.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine the rest mass of the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\Lambda ^0}">
<mrow>
<msup>
<mi mathvariant="normal">Λ</mi>
<mn>0</mn>
</msup>
</mrow>
</math></span> particle.</p>
<div class="marks">[4]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Determine, using your answer to (a), the initial speed of the <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\Lambda ^0}">
<mrow>
<msup>
<mi mathvariant="normal">Λ</mi>
<mn>0</mn>
</msup>
</mrow>
</math></span> particle.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\Lambda ">
<mi mathvariant="normal">Λ</mi>
</math></span> momentum = 900</p>
<p><em>E</em><sub>proton</sub> = «<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {p{c^2} + {{\left( {m{c^2}} \right)}^2}} = \sqrt {{{630}^2} + {{938}^2}} = ">
<msqrt>
<mi>p</mi>
<mrow>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>m</mi>
<mrow>
<msup>
<mi>c</mi>
<mn>2</mn>
</msup>
</mrow>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mo>=</mo>
<msqrt>
<mrow>
<msup>
<mrow>
<mn>630</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mn>938</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mo>=</mo>
</math></span>» 1130 «MeV»</p>
<p><em>E</em><sub>pion</sub> = «<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {{{270}^2} + {{140}^2}} = ">
<msqrt>
<mrow>
<msup>
<mrow>
<mn>270</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>+</mo>
<mrow>
<msup>
<mrow>
<mn>140</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mo>=</mo>
</math></span>» 304 «MeV»</p>
<p>so rest mass of <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\Lambda ">
<mi mathvariant="normal">Λ</mi>
</math></span> = «<span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\sqrt {{{\left( {1130 + 304} \right)}^2} - {{900}^2}} = ">
<msqrt>
<mrow>
<msup>
<mrow>
<mrow>
<mo>(</mo>
<mrow>
<mn>1130</mn>
<mo>+</mo>
<mn>304</mn>
</mrow>
<mo>)</mo>
</mrow>
</mrow>
<mn>2</mn>
</msup>
</mrow>
<mo>−</mo>
<mrow>
<msup>
<mrow>
<mn>900</mn>
</mrow>
<mn>2</mn>
</msup>
</mrow>
</msqrt>
<mo>=</mo>
</math></span>» 1116 «MeV c<sup>–2</sup>»</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>«<em>E = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\gamma ">
<mi>γ</mi>
</math></span> mc<sup>2</sup></em> so» <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\gamma ">
<mi>γ</mi>
</math></span> = « <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{1434}}{{1116}}">
<mfrac>
<mrow>
<mn>1434</mn>
</mrow>
<mrow>
<mn>1116</mn>
</mrow>
</mfrac>
</math></span> =» 1.28</p>
<p>to give 0.64<em>c</em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="background-color: #ffffff;">A Σ<sup>+</sup> particle decays from rest into a neutron and another particle X according to the reaction</span></p>
<p style="text-align: center;"><span style="background-color: #ffffff;">Σ<sup>+</sup> → n + X</span></p>
<p><span style="background-color: #ffffff;">The following data are available.</span></p>
<p style="padding-left: 90px;"><span style="background-color: #ffffff;">Rest mass of Σ<sup>+</sup> = 1190 MeV c<sup>–2</sup><br>Momentum of neutron = 185 MeV c<sup>–1</sup></span></p>
</div>
<div class="specification">
<p><span style="background-color: #ffffff;">Calculate, for the neutron,</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="background-color: #ffffff;">the total energy.</span></p>
<div class="marks">[1]</div>
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="background-color: #ffffff;">the speed.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="background-color: #ffffff;">Determine the rest mass of X.</span></p>
<div class="marks">[3]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="background-color: #ffffff;">neutron energy <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><msqrt><msup><mn>185</mn><mn>2</mn></msup><mo>+</mo><msup><mn>940</mn><mn>2</mn></msup></msqrt><mo>=</mo><mn>958</mn></math> «MeV» ✔<br></span></p>
<p><em><span style="background-color: #ffffff;">NOTE: Allow 1.5 × 10<sup>–10</sup> «J»</span></em></p>
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>ALTERNATIVE 1</strong></em></p>
<p><span style="background-color: #ffffff;">«use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mi>γ</mi><mo> </mo><msub><mi>E</mi><mn>0</mn></msub></math>»</span></p>
<p><span style="background-color: #ffffff;"><span style="background-color: #ffffff;">«<math xmlns="http://www.w3.org/1998/Math/MathML"><mn>958</mn><mo>=</mo><mn>940</mn><mi>γ</mi></math> so» <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>γ</mi><mo>=</mo><mn>1</mn><mo>.</mo><mn>019</mn></math> ✔</span></span></p>
<p><em><span style="background-color: #ffffff;"><span style="background-color: #ffffff;">v = </span></span></em><span style="background-color: #ffffff;"><span style="background-color: #ffffff;">0.193</span></span><em><span style="background-color: #ffffff;"><span style="background-color: #ffffff;">c </span></span></em><span style="background-color: #ffffff;"><span style="background-color: #ffffff;">✔</span></span></p>
<p> </p>
<p><em><strong><span style="background-color: #ffffff;"><span style="background-color: #ffffff;">ALTERNATIVE 2</span></span></strong></em></p>
<p><span style="background-color: #ffffff;"><span style="background-color: #ffffff;">«use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mi>γ</mi><mo> </mo><mi>m</mi><mi>v</mi></math>»</span></span></p>
<p><span style="background-color: #ffffff;"><span style="background-color: #ffffff;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>185</mn><mo>=</mo><mn>940</mn><mfrac><mstyle displaystyle="true"><mfrac><mi>v</mi><mi>c</mi></mfrac></mstyle><msqrt><mn>1</mn><mo>-</mo><msup><mfenced><mstyle displaystyle="true"><mfrac><mi>v</mi><mi>c</mi></mfrac></mstyle></mfenced><mn>2</mn></msup></msqrt></mfrac></math> ✔</span></span></p>
<p><span style="background-color: #ffffff;"><span style="background-color: #ffffff;"><em style="color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: italic;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;">v = </em>0.193<em style="color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: italic;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;">c </em>✔</span></span></p>
<p> </p>
<p><strong><em>ALTERNATIVE 3</em></strong></p>
<p>«use of <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>p</mi><mo>=</mo><mi>γ</mi><mo> </mo><mo> </mo><mi>m</mi><mi>v</mi></math>»</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mfrac><mrow><mi>p</mi><mi>c</mi></mrow><mi>E</mi></mfrac></math>✔</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><mfrac><mn>185</mn><mn>958</mn></mfrac><mo>=</mo><mn>0</mn><mo>.</mo><mn>193</mn><mi>c</mi></math> ✔</p>
<p style="color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"> </p>
<p style="color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"><em><span style="background-color: #ffffff;">NOTE: Allow v = 5.8 × 10<sup>7 </sup>«ms<sup>–1</sup>»</span></em> </p>
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>momentum of X = 185 <span style="background-color: #ffffff;">«MeV c<sup>–1</sup>»<span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;">✔</span></span></p>
<p><span style="background-color: #ffffff;">energy of X = 1190 – 958 = 232 <span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;">«MeV</span><span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;">»✔</span></span></p>
<p><span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>m</mi><mn>0</mn></msub><mo>=</mo><mo>«</mo><msqrt><msup><mn>232</mn><mn>2</mn></msup><mo>-</mo><msup><mn>185</mn><mn>2</mn></msup></msqrt><mo>=</mo><mo>»</mo><mn>140</mn></math> «MeV c</span><sup style="color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 11.6px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;">–2</sup><span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;">»</span><span style="background-color: #ffffff;"><span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"><span style="background-color: #ffffff;"> ✔</span></span></span></p>
<p><span style="font-size: 14px;"><em><span style="background-color: #ffffff;"><span style="text-align: left;color: #000000;text-indent: 0px;letter-spacing: normal;font-family: Verdana,Arial,Helvetica,sans-serif;font-variant: normal;font-weight: 400;text-decoration: none;display: inline !important;white-space: normal;float: none;background-color: #ffffff;"><span style="background-color: #ffffff;"> NOTE: Allow mass in kg - gives 2.5 × 10<sup>–28 </sup>«kg»</span></span></span></em></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a(i).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a(ii).</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p><span style="background-color: #ffffff;">A black hole has a Schwarzschild radius<em> R</em>. A probe at a distance of 0.5<em>R</em> from the event horizon of the black hole emits radio waves of frequency <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math> that are received by an observer very far from the black hole.</span></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="background-color: #ffffff;">Explain why the frequency of the radio waves detected by the observer is lower than <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi></math>.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p><span style="background-color: #ffffff;">The probe emits 20 short pulses of these radio waves every minute, according to a clock in the probe. Calculate the time between pulses as measured by the observer.</span></p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p><span style="background-color: #ffffff;"><em><strong>ALTERNATIVE 1</strong></em><br>as the photons move away from the black hole, they lose energy in the gravitational field ✔<br>since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><mi>h</mi><mi>f</mi></math> «the detected frequency is lower than the emitted frequency» ✔</span></p>
<p> </p>
<p><span style="background-color: #ffffff;"><em><strong>ALTERNATIVE 2</strong></em><br>if the observer was accelerating away from the probe, radio waves would undergo Doppler shift towards lower frequency ✔<br>by the equivalence principle, the gravitational field has the same effect as acceleration ✔</span></p>
<p> </p>
<p><span style="background-color: #ffffff;"><em><strong>ALTERNATIVE 3</strong></em><br>due to gravitational time dilation, time between arrivals of wavefronts is greater for the observer ✔<br>since <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mo>=</mo><mfrac><mn>1</mn><mi>T</mi></mfrac></math>, «the detected frequency is lower than the emitted frequency» ✔</span></p>
<p> </p>
<p><em><span style="background-color: #ffffff;">NOTE: The question states that received frequency is lower so take care not to award a mark for simply re-stating this, a valid explanation must be given.</span></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><span style="background-color: #ffffff;">time between pulses = 3 s according to the probe ✔</span></p>
<p><span style="background-color: #ffffff;"><span style="background-color: #ffffff;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∆</mo><mi>t</mi><mo>=</mo><mo>«</mo><mfrac><mn>3</mn><msqrt><mn>1</mn><mo>-</mo><mstyle displaystyle="true"><mfrac><mn>1</mn><mrow><mn>1</mn><mo>.</mo><mn>5</mn></mrow></mfrac></mstyle></msqrt></mfrac><mo>=</mo><mo>»</mo><mo> </mo><mn>5</mn><mo>.</mo><mn>2</mn></math> «s»<span style="display: inline !important;float: none;background-color: #ffffff;color: #000000;font-family: Verdana,Arial,Helvetica,sans-serif;font-size: 14px;font-style: normal;font-variant: normal;font-weight: 400;letter-spacing: normal;text-align: left;text-decoration: none;text-indent: 0px;white-space: normal;"> ✔</span></span></span></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>The Schwarzschild radius of a black hole is 6.0 x 10<sup>5</sup> m. A rocket is 7.0 x 10<sup>8</sup> m from the black hole and has a clock. The proper time interval between the ticks of the clock on the rocket is 1.0 s. These ticks are transmitted to a distant observer in a region free of gravitational fields.</p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Outline why the clock near the black hole runs slowly compared to a clock close to the distant observer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>Calculate the number of ticks detected in 10 ks by the distant observer.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>this is gravitational time dilation</p>
<p><em><strong>OR</strong></em></p>
<p>black hole gives rise to a «strong» gravitational field</p>
<p>clocks in stronger field run more slowly</p>
<p><em><strong>OR</strong></em></p>
<p>the clock «signal» is subject to gravitational red-shift</p>
<p>the clock is subject to gravitational red shift</p>
<p><em><strong>OR</strong></em></p>
<p>the clock has lost gravitational potential energy in moving close to the black hole</p>
<p><em><strong>[Max 2 Marks]</strong></em></p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>ALTERNATIVE 1 (10 ks is in observer frame):</strong></em></p>
<p>Δ<em>t'</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="10000\sqrt {1 - \frac{{6.0 \times {{10}^5}}}{{7.0 \times {{10}^8}}}} ">
<mn>10000</mn>
<msqrt>
<mn>1</mn>
<mo>−</mo>
<mfrac>
<mrow>
<mn>6.0</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mn>5</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>7.0</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mn>8</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</msqrt>
</math></span></p>
<p>9995.7 so 9995 «ticks»</p>
<p><em>Allow 9996</em></p>
<p><em>Allow ECF if 10 is used instead of 1000</em>0</p>
<p><em><strong>ALTERNATIVE 2 (10 ks is in rocket frame):</strong></em></p>
<p>Δ<em>t</em> = <span class="mjpage"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\frac{{10000}}{{\sqrt {1 - \frac{{6.0 \times {{10}^5}}}{{7.0 \times {{10}^8}}}} }}">
<mfrac>
<mrow>
<mn>10000</mn>
</mrow>
<mrow>
<msqrt>
<mn>1</mn>
<mo>−</mo>
<mfrac>
<mrow>
<mn>6.0</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mn>5</mn>
</msup>
</mrow>
</mrow>
<mrow>
<mn>7.0</mn>
<mo>×</mo>
<mrow>
<msup>
<mrow>
<mn>10</mn>
</mrow>
<mn>8</mn>
</msup>
</mrow>
</mrow>
</mfrac>
</msqrt>
</mrow>
</mfrac>
</math></span></p>
<p>10004 «ticks»</p>
<p><em>Allow ECF if 10 is used instead of 10000</em></p>
<div class="question_part_label">b.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
[N/A]
<div class="question_part_label">b.</div>
</div>
<br><hr><br><div class="specification">
<p>A box is in free fall in a uniform gravitational field. Observer X is at rest inside the box. Observer Y is at rest relative to the gravitational field. A light source inside the box emits a light ray that is initially parallel to the floor of the box according to both observers.</p>
<p style="text-align: center;"><img 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"></p>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State the equivalence principle.</p>
<div class="marks">[1]</div>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State and explain the path of the light ray according to observer X.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px; padding-right: 20px;">
<p>State and explain the path of the light ray according to observer Y.</p>
<div class="marks">[2]</div>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Markscheme</h2>
<div class="question" style="padding-left: 20px;">
<p>a freely falling frame in a gravitational field is equivalent to an inertial frame<br><em><strong>OR</strong></em><br>a frame accelerating in free space is equivalent to a frame at rest in a gravitational field ✔</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>X is in an inertial frame ✔</p>
<p>so light will follow a straight line path «parallel to the floor of the box» ✔</p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p><em><strong>ALTERNATIVE 1</strong></em><br>light must hit right wall of box at same place as determined by X ✔</p>
<p>«but box is accelerating» so path must be curved downward ✔</p>
<p><em><strong>ALTERNATIVE 2</strong></em><br>light is affected by gravity «for the observer at rest to the ground» ✔</p>
<p>so the path is curved downward/toward the ground ✔</p>
<div class="question_part_label">b.ii.</div>
</div>
<h2 style="margin-top: 1em">Examiners report</h2>
<div class="question" style="padding-left: 20px;">
<p>Freefall in a gravitational field. Most of the candidates answered well.</p>
<div class="question_part_label">a.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Freefall in a gravitational field. Most of the candidates answered well but in b) omitted to mention that X is inertial.</p>
<div class="question_part_label">b.i.</div>
</div>
<div class="question" style="padding-left: 20px;">
<p>Quite a high number of candidates, in b ii), stated incorrectly, that the light ray will be curved upward.</p>
<div class="question_part_label">b.ii.</div>
</div>
<br><hr><br>