A man weighing 80 kg is standing at the centre of a flat boat and he is 20 m from the shore. He walks 8 m on the boat towards the shore and then halts.The boat weight 200 kg. How far is he from the shore at the end of this time ?

Options

A. 11.2 m
B. 13.8 m
C. 14.3 m
D. 15.4 m

Answer

To solve the problem step by step, we will use the concept of the center of mass and the principle of conservation of momentum.

### Step 1: Understand the Initial Setup
- The man weighs 80 kg and is standing at the center of a flat boat weighing 200 kg.
- The initial distance from the shore is 20 meters.

### Step 2: Determine the Initial Position of the Center of Mass
- The initial position of the center of mass (CM) of the system (man + boat) can be calculated using the formula:

\[ x_{CM} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]

Where:
- \( m_1 = 80 \, \text{kg} \) (mass of the man)
- \( m_2 = 200 \, \text{kg} \) (mass of the boat)
- \( x_1 = 20 \, \text{m} \) (initial position of the man)
- \( x_2 = 20 \, \text{m} \) (initial position of the boat)

Plugging in the values:

\[ x_{CM} = \frac{80 \times 20 + 200 \times 20}{80 + 200} = \frac{1600 + 4000}{280} = \frac{5600}{280} = 20 \, \text{m} \]

### Step 3: Calculate the New Position After the Man Walks
- The man walks 8 m towards the shore. Thus, his new position relative to the boat is:

\[ x_{man} = 20 - 8 = 12 \, \text{m} \]

- Since the center of mass must remain unchanged (still at 20 m), the boat must move in the opposite direction to keep the center of mass fixed.

### Step 4: Determine How Far the Boat Moves
- Let \( d \) be the distance the boat moves away from the shore. The new position of the boat's center of mass can be expressed as:

\[ x_{CM} = \frac{80 \times 12 + 200 \times (20 + d)}{280} \]

Setting this equal to the initial center of mass position (20 m):

\[ 20 = \frac{80 \times 12 + 200 \times (20 + d)}{280} \]

Simplifying:

\[ 20 \times 280 = 80 \times 12 + 200 \times (20 + d) \]

\[ 5600 = 960 + 4000 + 200d \]

\[ 5600 - 4960 = 200d \]

\[ 640 = 200d \]

\[ d = \frac{640}{200} = 3.2 \, \text{m} \]

### Step 5: Calculate the Final Position of the Man
- The final position of the man from the shore is:

\[ \text{Final position} = \text{Initial position} - \text{Distance moved by man} + \text{Distance moved by boat} \]

\[ = 20 - 8 + 3.2 = 15.2 \, \text{m} \]

Thus, the man is **15.2 meters from the shore** at the end of this time.