A body of mass m moving with a velocity v is acted upon by a force. Write expression for change in momentum in each of the following cases (i) when v << c (ii) when v → c, and (iii) when v << c but m does not remain constant. Here c is the speed of light.

(i) When v << c

i.e., if the velocity of the moving particle is much smaller than the velocity of light (c) , then —

Change in momentum (Δp) = m Δv

It happens when the velocity of particle is of the order of 10⁶ m s^-1 or less than this, then the variation in mass with velocity is small enough and mass can be considered to be constant.

(ii) when v → c

Change in momentum (Δp) = Δ(mv)

(iii) when v << c but m does not remain constant.

In case of atomic particles moving with velocity comparable to the velocity of light c, it was observed that mass of the particle does not remain constant, but it increases with increase in velocity, according to the relation

m = $\dfrac{m_0}{\sqrt {1-\big(\dfrac{v}{c}\big)^2}}$

where, m₀ is the mass of the particle when it is at rest (i.e., v = 0)

Then,

Change in momentum (Δp) = Δ(mv)