Question

A man of mass $M=50\text{kg}$ is standing at one end of a boat of mass $m=300\text{kg}$ floating on still water. He walks towards the other end of the boat with a constant velocity $(v=2\text{m/s})$ with respect to a stationary observer on land. Choose the correct options.

• A. Velocity of $COM$ will be zero
• B. $COM$ will move with constant velocity.
• C. The magnitude of relative velocity of the boat with respect to the walking man is greater than the magnitude of the relative velocity of the boat w.r.t. the stationary observer.
• D. The magnitude of relative velocity of the boat with respect to the walking man is less than the magnitude of the relative velocity of the boat w.r.t the stationary observer.

Answer 1

Answer: (A), (C)

The magnitude of relative velocity of the boat with respect to the walking man is greater than the magnitude of the relative velocity of the boat w.r.t. the stationary observer.

Given,
mass of man $M=\left(m_1\right)=50\text{kg}$
mass of boat $m=\left(m_2\right)=300\text{kg}$

With respect to the stationary observer, man moves with a velocity $v=2\text{m/s}$ and assume boat moves with a velocity $=v_1\text{m/s}$.

As we know, there is no external force on the system. Hence, the velocity of $COM$ is zero $(V_{com}=0)$

To determine the velocity of the boat with respect to a stationary observer on land :-
$V_{com}=\frac{m_{1}v+m_2{v}_1}{m_1+m_2}=0$
$m_{1}v+m_2{v}_1=0$
$50\times 2+300\times v_1=0$
$v_1=\frac{-1}{3}\text{m/s}=-0.33\text{m/s}$

The negative sign implies that boat is moving in a direction opposite to that of the walking man on the boat (for a stationary observer on land)

To determine velocity of boat w.r.t walking man, we can find relative velocity as
$v_{bm}=v_b-v_m$
where $v_{bm}=$ velocity of boat w.r.t walking man
$v_b=$ velocity of boat w.r.t. stationary observer $=v_1$
$v_m=$ velocity of walking man w.r.t stationary observer $=v$
$\therefore v_{bm}=(-\frac{1}{3})-\left(2\right)=\frac{-7}{3}\text{m/s}$
$=-2.33\text{m/s}$

Hence magnitude of relative velocity of boat w.r.t the walking man is greater than the magnitude of the relative velocity of boat w.r.t the stationary observer.
Option (a) and (c) are correct.