A lambda Λ^0 particle at rest decays into a proton p and a pion π^- according to the reaction Λ^0 → p + π^- where the rest energy of p = 938 MeV and the rest energy of π^- = 140 MeV. The speed of the pion after the decay is 0.579c. For this speed γ = 1.2265.
Calculate the speed of the proton.
Muons are created at a height of 3230 m above the Earth's surface. The muons move vertically downward at a speed of 0.980 c relative to the Earth's surface. The gamma factor for this speed is 5.00. The half-life of a muon in its rest frame is 2.20 μs.
Estimate in the Earth frame the fraction of the original muons that will reach the Earth's surface before decaying according to Newtonian mechanics.
Estimate in the Earth frame the fraction of the original muons that will reach the Earth's surface before decaying according to special relativity.
Demonstrate how an observer moving with the same velocity as the muons accounts for the answer to (ii).
An astronaut is orbiting Earth in a spaceship. Why does the astronaut experience weightlessness?
This question is about the mass-radius relation for a certain type of star. The radius R and mass M of ten different stars were measured and the results are shown plotted below. The radius is expressed in terms of the Sun's radius R_S and the mass in terms of the Sun's mass M_S. The uncertainty in the measurement of the mass is negligible. The uncertainty in the measurement of the radius is ±0.05 R_S.
Draw error bars for the first and the last data points.
Suggest why there might be a linear relationship between R and M for these stars.
Determine the equation for this linear relationship.
Estimate the maximum value for the mass of this type of star.
Suggest why no star of this type can in fact have a mass equal to your answer to (iii).
Draw a line of best-fit for the data.
The new data suggests that the maximum value for the mass of this type of star is different from your answer in (iii). Estimate this new value.
Suggest why your answer to (ii) is only an estimate.
Explain how a graph may be used to verify this hypothesis.
Explain how a graph may be used to determine the constant n.
This question is about relativistic mechanics. A rho meson (ρ) decays at rest in a laboratory into a pion (π+) and an anti-pion (π−) according to ρ → π+ + π− The rest masses of the particles involved are: m_π+ = m_π− = 140 MeV c^−2 m_ρ = 770 MeV c^−2
Show that the initial momentum of the pion is 360 MeV c^−1.
Show that the speed of the pion relative to the laboratory is 0.932 c.
Calculate, in MeV c^−2, the mass that has been converted into energy in this decay.
The pion (π+) emits a muon in the same direction as the velocity of the pion. The speed of the muon is 0.271 c relative to the pion. Calculate the speed of the muon relative to the laboratory.
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 f 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 f
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
This question is about relativistic kinematics. The diagram shows a spaceship as it moves past Earth on its way to a planet P. The planet is at rest relative to Earth. The distance between the Earth and planet P is 12 ly as measured by observers on Earth. The spaceship moves with speed 0.60 c relative to Earth. Consider two events: Event 1: when the spaceship is above Earth Event 2: when the spaceship is above planet P Judy is in the spaceship and Peter is at rest on Earth.
State the reason why the time interval between event 1 and event 2 is a proper time interval as measured by Judy.
Calculate the time interval between event 1 and event 2 according to Peter.
Calculate the time interval between event 1 and event 2 according to Judy.
Judy considers herself to be at rest. According to Judy, the Earth and planet P are moving to the left.
Calculate, according to Judy, the distance separating the Earth and planet P.
Using your answers to (ii) and (c)(i), determine the speed of planet P relative to the spaceship.
Comment on your answer to (ii).
At a point half-way between the Earth and planet P, the spaceship passes a space station that is at rest relative to the Earth and planet P. At that instant, radio signals are sent towards the spaceship from the Earth and planet P. The signals are emitted simultaneously according to an observer S at rest on the space station.
Determine, according to Judy in the spaceship, which signal is emitted first.
On reaching planet P, the spaceship circles the planet and begins the return trip back to Earth. This situation leads to the twin paradox.
Describe what is meant by the term twin paradox.
Suggest how this paradox is resolved.
Outline what is meant by escape speed.
A probe is launched vertically upwards from the surface of a planet with a speed
v = ¾ v_esc
where v_esc is the escape speed from the planet. The planet has no atmosphere.
Determine, in terms of the radius of the planet R, the maximum height from the surface of the planet reached by the probe.
The total energy of a probe in orbit around a planet of mass M is E = -GMm/2r where m is the mass of the probe and r is the orbit radius. A probe in low orbit experiences a small frictional force. Suggest the effect of this force on the speed of the probe.
In an experiment a source of iron-57 emits gamma rays of energy 14.4 keV. 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.
This question is about black holes. A space probe is stationary in the gravitational field of a black hole. The mass of the black hole is . The space probe is emitting a pulse of blue light at a time interval of 1.0 seconds as measured on the space probe. The light is received by an observer on a distant space station that is stationary with respect to the space probe.
Explain why the light reaching the space station will be red-shifted.
The time between the pulses as measured by the observer on the distant space station is found to be 1.5 s . Calculate the distance of the space probe from the black hole.