Two balls A and B of masses m and 2m are in motion with velocities 2v and v respectively.
Compare
(i) their inertia, (ii) their momentum, and (iii) the force needed to stop them in same time.

(i) Given,

Mass of A = m

Mass of B = 2m

The factor on which inertia of a body depends is mass.

More the mass, more is the inertia of the body.

Therefore,

$\dfrac{\text {Inertia of A}}{\text {Inertia of B}} = \dfrac{\text {mass of A}}{\text {mass of B}} \\[0.5em]$

Substituting the values, we get,

$\dfrac{\text {Inertia of A}}{\text {Inertia of B}} = \dfrac{\text {m}}{\text {2m}} \\[0.5em] \dfrac{\text {Inertia of A}}{\text {Inertia of B}} = \dfrac{\text {1}}{\text {2}} \\[0.5em]$

Hence, ratio of their inertia = 1 : 2

(ii) As we know, momentum of a body (p) = mass (m) x velocity (v)

Given,

v_A = 2v

v_B = v

Ratio between the two is —

$\dfrac{{P}$A}{{P}B} = \dfrac{(mv)A}{(mv)B} \\[0.5em]

Substituting the values, we get,

$\dfrac{{P}$A}{{P}B} = \dfrac{m \times 2v}{2m \times v} \\[0.5em] \dfrac{{P}A}{{P}B} = \dfrac{2mv}{2mv} \\[0.5em] \Rightarrow \dfrac{{P}A}{{P}B} = \dfrac{1}{1} \\[0.5em]

Hence, ratio between the momentum of A and B is 1 : 1

(iii) According to Newton's second law of motion, the rate of change of momentum of a body is directly proportional to the force applied on it and as the ratio of momentum between A and B is 1 : 1, hence, ratio of force needed to stop A and B is also 1 : 1