Problem 3: An electron which has a mass of 9.11 × 10-31 kg, moves with a speed of 0.75 c. Find its relativistic momentum and compare this value with the momentum calculated from classical expression.

Solution

We know that momentum p = mv
When a body is moving with relativistic velocity (relativistic velocity is extremely large velocity, comparable with the velocity of light), its mass increases according to the relation

So, when mass is increased, the momentum of the relativistic body is also increased. That is, the increase in momentum is due to the increase in velocity as well as due to the increase in mass. In the given problem, we are calculating the momentum of the electron both with its relativistic mass and proper mass and then compare them to find what percentage of momentum is increased due increase in the mass.

Given

Rest mass of electron, m0 = 9.11 × 10-31 kg
Relativistic velocity, v = 0.75c
Speed of light, c = 3 × 108 m/s

Find

Rest mass momentum of electron,
pr Relativistic momentum of electron, p
% change in momentum

a: Rest mass momentum of electron, p = m0v. Put the values,

(b) Relativistic momentum, pr = mv. Here, m is the relativistic. We find the relativistic mass first by using formula (A).

We see the relativistic mass has increased as compared to the rest mass (9.11 × 10-31 kg). Now put this value of relativistic mass to find the relativistic momentum.

Compare with equation (1) and see the relativistic momentum has also increased.
Now increase in momentum = pr – p . Put the values from (1) and (2)
Increase in momentum = 31.05 × 10-23 – 20.49 × 10-23 = (31.05 – 20.9) × 10-23 = 10.56 × 10-23

c: Now to find % increase in the momentum
% increase in momentum = (increase in momentum/initial momentum) × 100