Question

When two identical balls $b_1$ and $b_2$, which have equal masses are moving with velocities, $V$ and $2V$ respectively, the ratio of inertia and momentum of $b_1$ to $b_2$ are:

• A. $2:1$ and $1:2$
• B. $1:1$ and $1:2$
• C. $1:1$ and $1:1$
• D. $1:1$ and $2:1$

Answer 1

The correct option is B $1:1$ and $1:2$

The inertia of a body depends on its mass.
As the balls have equal mass, the ball $b_1$ and $b_2$ have same inertia.
Let the mass of the balls be $M$.
$I_1$ and $I_2$ are the inertia of ball $b_1$ and $b_2$ respectively
$I_1$ = $K\times M$
$I_2$ = $K\times M$
where , $K$ is constant and $M$ is the mass of the ball.

Case 1: Ratio of inertia

$\frac{I_1}{1_2}=\frac{K\times M}{K\times M}$
$\Rightarrow$$\frac{I_1}{1_2}=\frac{1}{1}$

Case 2: Ratio of momentum

Momentum is the product of mass and velocity, so the ball with greater velocity has greater momentum.
Let $P_1$ and $P_2$ be the momentum of $b_1$ and $b_2$ respectively.
Let $V_1$ and $V_2$ be the velocities of $b_1$ and $b_2$.
Given $V_1$ = $V$ and $V_2$ = $2V$
$\frac{P_1}{P_2}=\frac{M\times V}{M\times 2V}$
$\Rightarrow$$\frac{P_1}{P_2}=\frac{1}{2}$.