Time dilation

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.

Figure 3 shows a modern version of Bertozzi’s experiment to measure the kinetic energy of high-speed electrons. A timer is used to measure the time taken for a pulse of electrons to travel from the detector to the aluminium block.

A potential difference (pd) of 1.30 MV is used to accelerate the electrons.

Show that each electron gains approximately 2 × 10⁻¹³ J of kinetic energy.

These electrons cause the temperature of the aluminium block to increase by 68.0 K The number of electrons that cause this increase in temperature is 4.50 × 10¹⁷

Deduce whether this increase in temperature is consistent with an accelerating pd of 1.30 MV.

specific heat capacity of aluminium = 903 J kg⁻¹ K⁻¹ mass of aluminium block = 1.50 kg

The speed of the electrons between the detector and the block is 2.88 × 10⁸ m s⁻¹.

Student A suggests that the non-relativistic equation for kinetic energy could be used. Student B suggests that the relativistic equation for kinetic energy is required in this situation.

Evaluate the suggestions of student A and student B. Support your answer with calculations.

The timer in Figure 3 records a time of 29.8 ns.

What is the proper time interval for an electron travelling from the detector to the aluminium block?

Tick (✓) one box.

The electrons in Figure 3 were accelerated from rest in 13 stages.

In each stage the electrons were accelerated by a pd of 100 kV.

As a result, an electron increases its speed and kinetic energy during each stage.

Compare, for an electron,

• its increase in speed for the first stage with that for the last stage

• its increase in kinetic energy for the first stage with that for the last stage.

Justify your answer.

No further calculations are required.

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.

A muon travels at a speed of 0.95 c relative to an observer.

The muon travels a distance of 2.5 × 10³ m between two points in the frame of reference of the observer.

Calculate the distance between these two points in the frame of reference of the muon.

Measurements of muons created by cosmic rays can be used to demonstrate relativistic time dilation.

State the measurements made and the observation that provides evidence for relativistic time dilation.

As the muons travel through the atmosphere, their speeds are reduced by interaction with the particles in the air.

Discuss, with reference to relativity, the effect that this reduction of speed has on the rate of detection of the muons on the surface of the Earth.

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.