Calculate the velocity of the spaceship relative to the Earth.

Explain what is meant by the statement that the spacetime interval is an invariant quantity.

Discuss the change in d according to observer Y.

Discuss how your result in (b)(i) and the outcome of the muon decay experiment support the theory of special relativity.

Explain whether or not the arrival times of the two flashes in the Earth frame are simultaneous events in the frame of rocket A.

Calculate the velocity of rocket B relative to rocket A.

Calculate the speed of the probe relative to the ground.

Determine the time it takes the probe to reach the front of the rocket according to an observer at rest on the ground.

ct′ = –1.1 m.

Calculate, for the reference frame of rocket A, the speed of rocket B according to the Lorentz transformation.

The spaceship passes the space station 90 minutes later as measured by the spaceship clock. Determine, for the reference frame of the Earth, the distance between the Earth and the space station.

determine the time between them according to observer B.

Demonstrate how an observer moving with the same velocity as the muons accounts for the answer to (a)(ii).

Show that the value of the invariant spacetime interval between events 1 and 2 is 9600 ly².

which lamp turns on first.

Calculate, according to the theory of special relativity, the time taken for a muon to reach the ground in the reference frame of the muon.

Explain how the force in part (b)(ii) arises.

Calculate, for observer A, the length L_A of the bridge

Determine, according to an observer in A, the velocity of B.

Later the train is travelling at a speed of 0.60c. Observer P measures the length of the train to be 125 m. The train enters a tunnel of length 100 m according to observer Q.

Show that the length of the train according to observer Q is 100 m.

A second train is moving at a velocity of –0.70c with respect to the ground.

Calculate the speed of the second train relative to observer P.

A charged pion decays spontaneously in a time of 26 ns as measured in the frame of reference in which it is stationary. The pion moves with a velocity of 0.96c relative to the Earth. Calculate the pion’s lifetime as measured by an observer on the Earth.

State which of the two time intervals is a proper time.

Calculate the length of the rocket according to X.

The velocity of P is 0.30c relative to the laboratory. A second particle Q moves at a velocity of 0.80c relative to the laboratory.

Calculate the speed of Q relative to P.

Determine the time, according to observer A, between X and Y.

As the spaceship passes the space station, the space station sends a radio signal back to the Earth. The reception of this signal at the Earth is event A. Determine the time on the Earth clock when event A occurs.

calculate the spacetime interval.

Outline why the observed times are different for A and B.

Estimate in the Earth frame the fraction of the original muons that will reach the Earth’s surface before decaying according to special relativity.

x′ = 1.5 m.

Calculate, for observer A, the time taken to cross the bridge.

Explain which of the lengths is the proper length.

Calculate the speed of the probe in terms of $c$, relative to Earth.

Calculate the speed of the probe in terms of $c$, relative to Earth.

Deduce the length of the probe as measured by an observer in the spaceship.

Determine, according to an observer in A, the time taken for B to meet A.

Identify the terms in the formula.

u′ = $\frac{u-v}{1-\frac{uv}{c^2}}$

Calculate the value of c ²t ² – x ².

Calculate the speed v of the train for the ratio $\frac{\mathrm{\Delta }{t}_{\text{P}}}{\mathrm{\Delta }{t}_{\text{Q}}}=0.30$.

Apply a Lorentz transformation to show that the time difference between events B and F according to observer P is 2.5 × 10^–7 s.

the time interval between the lamps turning on.

Demonstrate that the spacetime interval between events B and F is invariant.

Muons are unstable particles with a proper lifetime of 2.2 μs. Muons are produced 2.0 km above ground and move downwards at a speed of 0.98c relative to the ground. For this speed $\gamma$ = 5.0. Discuss, with suitable calculations, how this experiment provides evidence for time dilation.

A space shuttle is released from the rocket. The shuttle moves with speed 0.20c to the right according to X. Calculate the velocity of the shuttle relative to the rocket.