Relativistic mass & energy

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.

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.

Bertozzi investigated how the kinetic energy of electrons varies with speed.

Which graph shows the variation of kinetic energy with speed? Tick () one box.

Calculate the speed of a particle when its kinetic energy is equal to its rest energy.

speed = m s⁻¹

Discuss the change in the observed mass of a spring when it is stretched.

Table 1 shows data of speed v and kinetic energy E_k for electrons from a modern version of the Bertozzi experiment.

Classical mechanics predicts that E_k v².

Deduce whether the data in Table 1 are consistent with this prediction.

Discuss how Einstein’s theory of special relativity explains the data in Table 1.

Calculate, in J, the kinetic energy of one electron travelling at a speed of 0.95 c.

Calculate the velocity of the protons as a fraction of the speed of light.