Laws of Motion — Question 4

(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 —

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

Substituting the values, we get,

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

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