A truck of mass 10 metric ton runs at `3ms^(-1)` along a level track and collides with a loaded truck of mass 20 metric ton, standing at rest. If the trucks couple together , the common speed after the collision is

Options

A. `1 ms ^(-1)`
B. `0.1 ms ^(-1)`
C. `0.5 ms ^(-1)`
D. `0.3 ms ^(-1)`

Answer

To solve the problem step by step, we will use the principle of conservation of momentum. Here's the detailed solution:

### Step 1: Identify the masses and velocities before the collision
- Mass of the first truck (m1) = 10 metric tons = 10,000 kg (since 1 metric ton = 1,000 kg)
- Velocity of the first truck (u1) = 3 m/s
- Mass of the second truck (m2) = 20 metric tons = 20,000 kg
- Velocity of the second truck (u2) = 0 m/s (since it is at rest)

### Step 2: Calculate the initial momentum of the system
The initial momentum (P_initial) of the system can be calculated using the formula:
\[ P_{\text{initial}} = m_1 \cdot u_1 + m_2 \cdot u_2 \]

Substituting the values:
\[ P_{\text{initial}} = (10,000 \, \text{kg} \cdot 3 \, \text{m/s}) + (20,000 \, \text{kg} \cdot 0 \, \text{m/s}) \]
\[ P_{\text{initial}} = 30,000 \, \text{kg m/s} + 0 \]
\[ P_{\text{initial}} = 30,000 \, \text{kg m/s} \]

### Step 3: Calculate the final momentum of the system
After the collision, both trucks couple together and move with a common velocity (V). The total mass after the collision is:
\[ m_{\text{total}} = m_1 + m_2 = 10,000 \, \text{kg} + 20,000 \, \text{kg} = 30,000 \, \text{kg} \]

The final momentum (P_final) of the system is given by:
\[ P_{\text{final}} = m_{\text{total}} \cdot V \]
\[ P_{\text{final}} = 30,000 \, \text{kg} \cdot V \]

### Step 4: Apply the conservation of momentum
According to the conservation of momentum:
\[ P_{\text{initial}} = P_{\text{final}} \]
\[ 30,000 \, \text{kg m/s} = 30,000 \, \text{kg} \cdot V \]

### Step 5: Solve for V
To find the common velocity (V), we can rearrange the equation:
\[ V = \frac{30,000 \, \text{kg m/s}}{30,000 \, \text{kg}} \]
\[ V = 1 \, \text{m/s} \]

### Conclusion
The common speed after the collision is **1 m/s**.