A monkey of mass 20 kg rides on a 40 kg trolley moving at a constant speed of 8 m/s along a horizontal track. Frictional force can be neglected. If the monkey jumps vertically, with respect to the moving trolley, the catch the overhanging branch of tree, the speed of the trolley after the monkey has jumped off is :-

Options

A. 8 m/s
B. 1 m/s
C. 4 m/s
D. 12 m/s

Answer

To solve the problem, we will use the principle of conservation of momentum. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the initial conditions
- The mass of the monkey (m1) = 20 kg
- The mass of the trolley (m2) = 40 kg
- The initial speed of the trolley (v_initial) = 8 m/s

### Step 2: Calculate the total initial momentum
The total initial momentum (p_initial) of the system (monkey + trolley) can be calculated using the formula:
\[ p_{\text{initial}} = (m_1 + m_2) \times v_{\text{initial}} \]
Substituting the values:
\[ p_{\text{initial}} = (20 \, \text{kg} + 40 \, \text{kg}) \times 8 \, \text{m/s } \]
\[ p_{\text{initial}} = 60 \, \text{kg} \times 8 \, \text{m/s} = 480 \, \text{kg m/s} \]

### Step 3: Analyze the final conditions after the monkey jumps
After the monkey jumps, the monkey will have a vertical velocity but no horizontal momentum (since it jumps straight up). The trolley will continue to move horizontally with a new speed (v_final).

### Step 4: Write the equation for final momentum
The final momentum (p_final) of the system can be expressed as:
\[ p_{\text{final}} = m_1 \times v_{monkey} + m_2 \times v_{trolley} \]
Since the monkey has no horizontal velocity after jumping, we can say:
\[ p_{\text{final}} = 0 + 40 \, \text{kg} \times v_{trolley} \]
So,
\[ p_{\text{final}} = 40 \, \text{kg} \times v_{trolley} \]

### Step 5: Apply conservation of momentum
According to the conservation of momentum:
\[ p_{\text{initial}} = p_{\text{final}} \]
Thus,
\[ 480 \, \text{kg m/s} = 40 \, \text{kg} \times v_{trolley} \]

### Step 6: Solve for the final speed of the trolley
Now, we can solve for \( v_{trolley} \):
\[ v_{trolley} = \frac{480 \, \text{kg m/s}}{40 \, \text{kg}} \]
\[ v_{trolley} = 12 \, \text{m/s} \]

### Conclusion
The speed of the trolley after the monkey has jumped off is **12 m/s**.

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