A heavy vehicle moving with velocity 15 m/s strikes an object of very small mass at rest head on elastically. Velocity of object ater collision is

Options

A. 15 m/s
B. 25 m/s
C. 20 m/s
D. 30 m/s

Answer

To solve the problem of a heavy vehicle colliding elastically with a small mass at rest, we will use the principles of conservation of momentum and the properties of elastic collisions.

### Step-by-Step Solution:

1. **Identify the Variables:**
- Let the mass of the heavy vehicle be \( M \).
- Let the mass of the small object be \( m \) (where \( m \) is very small compared to \( M \)).
- The initial velocity of the heavy vehicle \( u_1 = 15 \, \text{m/s} \).
- The initial velocity of the small object \( u_2 = 0 \, \text{m/s} \).
- The final velocity of the heavy vehicle after the collision \( v_1 \) (unknown).
- The final velocity of the small object after the collision \( v_2 \) (unknown).

2. **Apply Conservation of Momentum:**
The total momentum before the collision equals the total momentum after the collision.
\[ Mu_1 + mu_2 = Mv_1 + mv_2 \]
Substituting the known values:
\[ M(15) + m(0) = Mv_1 + mv_2 \]
This simplifies to:
\[ 15M = Mv_1 + mv_2 \quad \text{(1)} \]

3. **Apply the Coefficient of Restitution:**
For an elastic collision, the coefficient of restitution \( e = 1 \). This means:
\[ e = \frac{v_2 - v_1}{u_1 - u_2} \]
Substituting the known values:
\[ 1 = \frac{v_2 - v_1}{15 - 0} \]
This simplifies to:
\[ v_2 - v_1 = 15 \quad \text{(2)} \]

4. **Substitute Equation (2) into Equation (1):**
From equation (2), we can express \( v_1 \) in terms of \( v_2 \):
\[ v_1 = v_2 - 15 \]
Substitute this into equation (1):
\[ 15M = M(v_2 - 15) + mv_2 \]
Expanding this gives:
\[ 15M = Mv_2 - 15M + mv_2 \]
Rearranging terms:
\[ 30M = Mv_2 + mv_2 \]
Factoring out \( v_2 \):
\[ 30M = v_2(M + m) \quad \text{(3)} \]

5. **Neglect the Small Mass \( m \):**
Since \( m \) is very small compared to \( M \), we can neglect it in the equation:
\[ 30M \approx v_2 M \]
Thus:
\[ v_2 \approx \frac{30M}{M} = 30 \, \text{m/s} \]

### Conclusion:
The velocity of the small object after the collision is approximately \( 30 \, \text{m/s} \).