Question

A man of mass $m$ walks from end $A$ to the other end $B$ of a boat of mass $M$ and length $l$. The coefficient of friction between the man and the boat is $\mu$ and neglect any resistive force between the boat and the water.

• A. If the man runs at his maximum acceleration, the acceleration of the boat is $\left(m/M\right)g$.
• B. The maximum time take by the men to reach the other end of the boat is $\sqrt{\frac{2Ml}{(M+m)\mu g}}$.
• C. If the man runs at his maximum acceleration, the acceleration of the boat is $\frac{m}{m+M}\mu g$.
• D. The minimum time take by the men to reach the other end of the boat is $\sqrt{\frac{2ml}{(M+m)\mu g}}$.

Answer 1

Answer: (A), (B)

The correct options are
A If the man runs at his maximum acceleration, the acceleration of the boat is $\left(m/M\right)g$.
B The maximum time take by the men to reach the other end of the boat is $\sqrt{\frac{2Ml}{(M+m)\mu g}}$.

Maximum acceleration of the man $=\mu mg/m=\mu g$
Maximum acceleration of the boat $=\mu mg/M$
Maximum relative acceleration:
$a_{rel}=\mu g+\mu g\frac{m}{M}=\frac{(M+m)}{M}\mu g$
$S_{rel}=\frac{1}{2}{a}_{rel}{t}^2\Rightarrow t=\sqrt{\frac{2lM}{\mu g(M+m)}}$