A boat of mass `40kg` is at rest. A dog of mass `4kg` moves in the boat with a velocity of `10m//s`. What is the velocity of boat (nearly)?

Options

A. `4m//s`
B. `2m//s`
C. `8m//s`
D. `1 m//s`

Explanation

To solve the problem, we will use the principle of conservation of momentum. The total momentum of the system (boat + dog) before the dog starts moving must equal the total momentum after the dog starts moving.

### Step-by-Step Solution:

1. **Identify the masses:**
- Mass of the boat (M) = 40 kg
- Mass of the dog (m) = 4 kg

2. **Initial conditions:**
- The boat is initially at rest, so its initial velocity (V_boat_initial) = 0 m/s.
- The dog moves with a velocity (V_dog) = 10 m/s relative to the boat.

3. **Final conditions:**
- Let the final velocity of the boat be V_boat_final (which we need to find).

4. **Apply the conservation of momentum:**
- The total initial momentum of the system is:
\[ P_{initial} = (M \cdot V_{boat\_initial}) + (m \cdot V_{dog\_initial}) = (40 \cdot 0) + (4 \cdot 0) = 0 \]
- The total final momentum of the system is:
\[ P_{final} = (M \cdot V_{boat\_final}) + (m \cdot V_{dog\_final}) \]
- Since the dog moves at 10 m/s relative to the boat, the final velocity of the dog (V_dog_final) can be expressed as:
\[ V_{dog\_final} = V_{boat\_final} + 10 \]

5. **Set up the equation using conservation of momentum:**
\[ 0 = (M \cdot V_{boat\_final}) + (m \cdot (V_{boat\_final} + 10)) \]
Substituting the values of M and m:
\[ 0 = (40 \cdot V_{boat\_final}) + (4 \cdot (V_{boat\_final} + 10)) \]

6. **Expand and simplify the equation:**
\[ 0 = 40 V_{boat\_final} + 4 V_{boat\_final} + 40 \]
\[ 0 = 44 V_{boat\_final} + 40 \]

7. **Solve for V_boat_final:**
\[ 44 V_{boat\_final} = -40 \]
\[ V_{boat\_final} = -\frac{40}{44} = -\frac{10}{11} \approx -0.91 \text{ m/s} \]

8. **Interpret the result:**
The negative sign indicates that the boat moves in the opposite direction to the dog's movement. Thus, the velocity of the boat is approximately 0.91 m/s in the opposite direction.

### Final Answer:
The velocity of the boat is approximately **0.91 m/s** in the direction opposite to that of the dog's movement.

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