HL Paper 3
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.
Explain, in terms of a conservation law, why two photons need to be created.
Determine the speed of the incoming electron.
Calculate the energy and the momentum for each photon after the collision.
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.
The gravitational field strength at 20 Mm above the surface of the Earth is about 0.6 N kg–1. Estimate the time correction per day needed to the time signals, due to the gravitational redshift.
Suggest, whether your answer to (a) underestimates or overestimates the correction required to the time signal.
State what is meant by the event horizon of a black hole.
Show that the surface area A of the sphere corresponding to the event horizon is given by
.
Suggest why the surface area of the event horizon can never decrease.
The diagram shows a box that is falling freely in the gravitational field of a planet.
A photon of frequency f 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.
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.
Calculate the expected shift in frequency between the emitted and the detected gamma rays.
Explain whether the detected frequency would be greater or less than the emitted frequency.
A deuterium nucleus (rest mass ) is accelerated by a potential difference of .
Define rest mass.
Calculate the total energy of the deuterium particle in .
In relativistic reactions the mass of the products may be less than the mass of the reactants. Suggest what happens to the missing mass.
An observer A is on the surface of planet X. Observer B is in a stationary spaceship above the surface of planet X.
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 Δf.
Observer B then sends a signal with frequency 1500 THz towards observer A.
Calculate the shift in frequency observed by A in terms of Δf.
Calculate the gravitational field strength on the surface of planet X.
The following data is given:
Δf = 170 Hz.
The distance between observer A and B is 10 km.
Observer A now sends a beam of light initially parallel to the surface of the planet.
Explain why the path of the light is curved.
In the Pound–Rebka–Snider experiment, a source of gamma rays was placed vertically above a gamma ray detector, in a tower on Earth.
Calculate the fractional change in frequency of the gamma rays at the detector.
Explain the cause of the frequency shift for the gamma rays in your answer in (a) in the Earth’s gravitational field.
Explain the cause of the frequency shift for the gamma rays in your answer in (a) if the tower and detector were accelerating towards the gamma rays in free space.
Two protons, travelling in opposite directions, collide. Each has a total energy of 3.35 GeV.
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–2. The total energy of the emitted proton and neutron from the interaction is 6.20 GeV.
Calculate the gamma (γ) factor for one of the protons.
Determine, in terms of MeV c–1, the momentum of the pion.
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.
A positive pion decays into a positive muon and a neutrino.
The momentum of the muon is measured to be 29.8 MeV c–1 in a laboratory reference frame in which the pion is at rest. The rest mass of the muon is 105.7 MeV c–2 and the mass of the neutrino can be assumed to be zero.
For the laboratory reference frame
write down the momentum of the neutrino.
show that the energy of the pion is about 140 MeV.
State the rest mass of the pion with an appropriate unit.
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.
Outline what is meant by the event horizon of a black hole.
Calculate the distance of the event horizon of the black hole from its centre.
Mass of Sun = 2 × 1030 kg
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.
Discuss how the time for the periodic event as measured by an observer on the Earth changes with the orbital position of S-2.
A proton has a total energy 1050 MeV after being accelerated from rest through a potential difference V.
Define total energy.
Determine the momentum of the proton.
Determine the speed of the proton.
Calculate the potential difference V.
The particle omega minus () decays at rest into a neutral pion () and the xi baryon () according to
The pion momentum is 289.7 MeV c–1.
The rest masses of the particles are:
: 1672 MeV c–2
: 135.0 MeV c–2
: 1321 MeV c–2
Show that energy is conserved in this decay.
Calculate the speed of the pion.
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.
The momenta of the photons produced have magnitudes p1 and p2. A student writes the following correct equations.
p1 – p2 = 1.41 MeV c–1
p1 + p2 = 2.01 MeV c–1
Show that the momentum of the electron is 1.41 MeV c–1.
Explain the origin of each equation.
Calculate, in MeV c–1, p1 and p2.
A probe launched from a spacecraft moves towards the event horizon of a black hole.
State what is meant by the event horizon of a black hole.
The mass of the black hole is 4.0 × 1036 kg. Calculate the Schwarzschild radius of the black hole.
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.
A proton is accelerated from rest through a potential difference V to a speed of 0.86c.
Calculate the potential difference V.
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.
Determine the momentum of one of the photons.
A rocket is accelerating upwards at 9.8 m s-2 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.
Explain why a change in frequency is expected for the photon detected at the top of the rocket.
Calculate the frequency change.
A lambda 0 particle at rest decays into a proton p and a pion according to the reaction
0 → p + –
where the rest energy of p = 938 MeV and the rest energy of – = 140 MeV.
The speed of the pion after the decay is 0.579c. For this speed = 1.2265. Calculate the speed of the proton.
The (Lambda) particle decays spontaneously into a proton and a negatively charged pion of rest mass 140 MeV c–2. After the decay, the particles are moving in the same direction with a proton momentum of 630 MeV c–1 and a pion momentum of 270 MeV c–1.
Determine the rest mass of the particle.
Determine, using your answer to (a), the initial speed of the particle.
A Σ+ particle decays from rest into a neutron and another particle X according to the reaction
Σ+ → n + X
The following data are available.
Rest mass of Σ+ = 1190 MeV c–2
Momentum of neutron = 185 MeV c–1
Calculate, for the neutron,
the total energy.
the speed.
Determine the rest mass of X.
A black hole has a Schwarzschild radius R. A probe at a distance of 0.5R from the event horizon of the black hole emits radio waves of frequency that are received by an observer very far from the black hole.
Explain why the frequency of the radio waves detected by the observer is lower than .
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.
The Schwarzschild radius of a black hole is 6.0 x 105 m. A rocket is 7.0 x 108 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.
Outline why the clock near the black hole runs slowly compared to a clock close to the distant observer.
Calculate the number of ticks detected in 10 ks by the distant observer.
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.
State the equivalence principle.
State and explain the path of the light ray according to observer X.
State and explain the path of the light ray according to observer Y.