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` an neglect any resistive force between the boat and the water.

Options

A. (a)If the man runs at his maximum acceleration, the acceleration of the boat is `(m//M)mug`.
B. (b)The minimum time take by the man to reach the other end of the boat is `sqrt((2MI)/((M+m)mug))`
C. (c)If man runs at his maximum acceleration the acceleration of boat is `m/(m+M)mug`
D. (d)The minimum time take by the man to reach the other end of the boat is `sqrt((2ml)/((M+m)mug))`

Explanation

To solve the problem of a man walking from one end of a boat to the other, we need to analyze the forces and accelerations involved. Here's a step-by-step breakdown of the solution:

### Step 1: Understand the system
- We have a man with mass \( m \) walking on a boat with mass \( M \) and length \( l \).
- The coefficient of friction between the man and the boat is \( \mu \).
- We will neglect any resistive forces between the boat and the water.

### Step 2: Determine the maximum acceleration of the man
- The maximum force of friction that can act on the man as he walks is given by:
\[ F_{\text{friction}} = \mu \cdot m \cdot g \]
- This frictional force provides the maximum acceleration \( a_m \) of the man:
\[ a_m = \frac{F_{\text{friction}}}{m} = \frac{\mu \cdot m \cdot g}{m} = \mu \cdot g \]

### Step 3: Determine the acceleration of the boat
- According to Newton's third law, the man walking forward will exert a backward force on the boat. The acceleration \( a_b \) of the boat can be derived from the same frictional force:
\[ F_{\text{friction}} = M \cdot a_b \]
- Thus, the acceleration of the boat is:
\[ a_b = \frac{F_{\text{friction}}}{M} = \frac{\mu \cdot m \cdot g}{M} \]

### Step 4: Determine the relative acceleration
- The relative acceleration \( a_{\text{relative}} \) between the man and the boat is given by:
\[ a_{\text{relative}} = a_m + a_b = \mu g + \frac{\mu m g}{M} \]
- This can be simplified to:
\[ a_{\text{relative}} = \mu g \left(1 + \frac{m}{M}\right) \]

### Step 5: Calculate the time taken to reach the other end of the boat
- The distance \( S \) that the man needs to cover is the length of the boat \( l \).
- Using the equation of motion, \( S = \frac{1}{2} a_{\text{relative}} t^2 \), we can solve for time \( t \):
\[ l = \frac{1}{2} a_{\text{relative}} t^2 \]
\[ t^2 = \frac{2l}{a_{\text{relative}}} \]
\[ t = \sqrt{\frac{2l}{\mu g \left(1 + \frac{m}{M}\right)}} \]

### Conclusion
- The maximum acceleration of the man is \( \mu g \).
- The maximum acceleration of the boat is \( \frac{\mu m g}{M} \).
- The time taken by the man to reach the other end of the boat is \( \sqrt{\frac{2l}{\mu g \left(1 + \frac{m}{M}\right)}} \).