A body of mass `0.25` kg is projected with muzzle velocity `100ms^(-1)` from a tank of mass 100 kg. What is the recoil velocity of the tank

Options

A. `5 ms^(-1)`
B. `25 ms^(-1)`
C. `0.5 ms^(-1)`
D. `0.25 ms^(-1)`

Explanation

To solve the problem of finding the recoil velocity of the tank when a body is projected from it, we will use the principle of conservation of momentum. Here’s a step-by-step solution:

### Step 1: Understand the system
We have a tank of mass \( m_t = 100 \, \text{kg} \) and a body of mass \( m_b = 0.25 \, \text{kg} \) that is projected with a muzzle velocity \( v_b = 100 \, \text{m/s} \). We need to find the recoil velocity \( v_t \) of the tank.

**Hint:** Identify the masses involved and the direction of motion for both the tank and the body.

### Step 2: Apply the conservation of momentum
According to the conservation of momentum, the total momentum before the projection must equal the total momentum after the projection. Initially, both the tank and the body are at rest, so the initial momentum \( P_{initial} = 0 \).

**Hint:** Remember that momentum is the product of mass and velocity.

### Step 3: Set up the equation
After the body is projected, the momentum of the system can be expressed as:
\[ P_{final} = m_b \cdot v_b + m_t \cdot (-v_t) \]
Here, \( v_t \) is the recoil velocity of the tank, and we take it as negative because it moves in the opposite direction to the body.

### Step 4: Write the equation based on conservation of momentum
Setting the initial momentum equal to the final momentum gives us:
\[ 0 = m_b \cdot v_b - m_t \cdot v_t \]
Substituting the known values:
\[ 0 = (0.25 \, \text{kg}) \cdot (100 \, \text{m/s}) - (100 \, \text{kg}) \cdot v_t \]

### Step 5: Solve for the recoil velocity \( v_t \)
Rearranging the equation:
\[ (100 \, \text{kg}) \cdot v_t = (0.25 \, \text{kg}) \cdot (100 \, \text{m/s}) \]
\[ v_t = \frac{(0.25 \, \text{kg}) \cdot (100 \, \text{m/s})}{100 \, \text{kg}} \]
\[ v_t = \frac{25}{100} = 0.25 \, \text{m/s} \]

### Final Answer
The recoil velocity of the tank is \( v_t = 0.25 \, \text{m/s} \).

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