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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 (m/M)g.
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B
The maximum time take by the men to reach the other end of the boat is 2Ml(M+m)μg.
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C
If the man runs at his maximum acceleration, the acceleration of the boat is mm+Mμg.
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D
The minimum time take by the men to reach the other end of the boat is 2ml(M+m)μg.
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Solution

The correct options are
A If the man runs at his maximum acceleration, the acceleration of the boat is $$ (m/M)g $$.
B The maximum time take by the men to reach the other end of the boat is $$ \sqrt { \dfrac { 2Ml }{ (M+m)\mu g } } $$.
Maximum acceleration of the man $$=\mu mg/m=\mu g$$
Maximum acceleration of the boat $$=\mu mg/M$$
Maximum relative acceleration:
$$a_{rel}=\mu g+\mu g\dfrac{m}{M}=\dfrac{(M+m)}{M}\mu g$$
$$S_{rel}=\dfrac{1}{2}a_{rel}t^2\Rightarrow t=\sqrt{\dfrac{2lM}{\mu g (M+m)}}$$

Physics

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