Special relativity

In a science-fiction novel, spaceship A transports cargo from planet O to planet P.

A travels at a constant speed of 0.50 c in the reference frame of an observer on P.

The observer on P measures the distance from O to P as 1.4 × 10¹⁴ m. Some of the cargo is food and must experience a journey time of no more than 10 days.

Determine whether A is able to deliver the cargo within this time.

The observer on P measures the length of a cargo at rest on P to be 210 m. There is a cargo hold along the full length of A. The observer on P measures the length of A to be 200 m in the direction of travel while the spaceship is moving at 0.50 c.

Determine whether the cargo will fit into the cargo hold of A.

Spaceships A and B are travelling towards each other along the same straight line. The speed of B is 0.30 c with respect to P.

Figure 5 shows the situation.

B sends a radio signal to A.

What is the speed of the radio signal measured by A ?

Tick (✓) one box.

Einstein developed his theory of special relativity from two postulates. One postulate states that physical laws have the same form in all inertial frames.

State the other postulate and explain how it is consistent with the equation:

A proton leaves a particle accelerator at point X and moves at a constant speed towards a target at point Y. The speed of the proton is 2.5 × 10⁸ m s⁻¹ in the frame of reference of the target.

The distance XY in the frame of reference of the proton is 38 m.

Calculate the distance XY in the frame of reference of the target.

Show that the kinetic energy E_k of the proton is about 1.2 × 10⁻¹⁰ J.

Sketch on Figure 4 the variation of E_k with speed v for a proton.

To help you, the dashed line represents

where m₀ is equal to the mass of a proton at rest.

State what is meant by an inertial frame of reference.

A pair of detectors is set up to measure the intensity of a parallel beam of unstable particles. In the reference frame of the laboratory, the detectors are separated by a distance of 45 m. The speed of the particles in the beam is 0.97 c.

The intensity of the beam at the second detector is 12.5% of the intensity at the first detector.

Calculate the half-life of the particles in the reference frame in which they are at rest.

In calculations involving time dilation, it is important to identify proper time.

Identify the proper time in the calculation in Question 04.2.

Figure 6 shows a diagram of the Michelson-Morley interferometer that was used to try to detect the absolute motion of the Earth through the ether (æther).

Light from the monochromatic source passes through the semi-silvered glass block and takes two different paths to the viewing telescope. The two paths, PM₁ and PM₂ are the same length. Interference fringes are observed through the viewing telescope.

It was predicted that when the interferometer was rotated through 90^o the fringe pattern would shift by 0.4 of the fringe spacing.

Explain how the experiment provided a means of testing the idea that the Earth had an absolute motion relative to the ether.

Your answer should include:

• an explanation of why a shift of the fringe pattern was predicted

• a comparison of the results of the experiment to the prediction

• the conclusion about the Earth’s absolute motion through the ether.

Extra space is available on the next page

The Michelson-Morley experiment provides evidence for one of the postulates of Einstein’s theory of special relativity.

State this postulate.

State the other postulate of Einstein’s theory of special relativity.

One consequence of the special theory of relativity is length contraction.

Experimental evidence for length contraction is provided by the decay of muons produced in the atmosphere by cosmic rays.

Figure 7 shows how the percentage of the number of muons remaining in a sample changes with time as measured by an observer in a frame of reference that is stationary relative to the muons.

In a particular experiment, muons moving with a velocity 0.990 c travel a distance of

1310 m through the atmosphere to a detector.

Determine the percentage of muons that reach the detector.

A student models a spacecraft journey that takes one year. The spacecraft travels directly away from an observer at a speed of 1.2 × 10⁷ m s⁻¹. The student predicts that a clock stationary relative to the observer will record a time several days longer than an identical clock on the spacecraft.

Comment on the student’s prediction. Support your answer with a time dilation calculation.

In practice, the gravitational field of the Sun affects the motion of the spacecraft and it does not travel directly away from the Earth throughout the journey.

Explain why this means that the theory of special relativity cannot be applied to the journey.