A heavy vehicle moving with velocity 15 m/s strikes an object of very small mass at rest head on elastically. Velocity of object after 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 can follow these steps:

### Step 1: Understand the Problem
We have a heavy vehicle (mass \( M \)) moving with a velocity of \( 15 \, \text{m/s} \) that collides head-on with a small object (mass \( m \)) at rest. We need to find the velocity of the small object after the collision.

### Step 2: Apply Conservation of Momentum
In an elastic collision, the total momentum before the collision is equal to the total momentum after the collision. The equation can be written as:

\[ M \cdot 15 + m \cdot 0 = M \cdot V_1 + m \cdot V_2 \]

Where:
- \( V_1 \) is the velocity of the heavy vehicle after the collision.
- \( V_2 \) is the velocity of the small object after the collision.

This simplifies to:

\[ 15M = M \cdot V_1 + m \cdot V_2 \tag{1} \]

### Step 3: Apply Conservation of Kinetic Energy
In an elastic collision, the total kinetic energy before the collision is equal to the total kinetic energy after the collision. The equation can be written as:

\[ \frac{1}{2} M (15)^2 + \frac{1}{2} m (0)^2 = \frac{1}{2} M V_1^2 + \frac{1}{2} m V_2^2 \]

This simplifies to:

\[ \frac{1}{2} M (15)^2 = \frac{1}{2} M V_1^2 + \frac{1}{2} m V_2^2 \tag{2} \]

### Step 4: Use the Velocity of Separation and Approach
For elastic collisions, the relative velocity of separation equals the relative velocity of approach:

\[ V_2 - V_1 = 15 \tag{3} \]

### Step 5: Solve the Equations
From equation (3), we can express \( V_1 \) in terms of \( V_2 \):

\[ V_1 = V_2 - 15 \tag{4} \]

Now substitute equation (4) into equation (1):

\[ 15M = M(V_2 - 15) + mV_2 \]

Expanding this gives:

\[ 15M = MV_2 - 15M + mV_2 \]

Rearranging terms:

\[ 15M + 15M = MV_2 + mV_2 \]

\[ 30M = (M + m)V_2 \]

### Step 6: Approximate for Large Mass
Since \( M \) is much larger than \( m \), we can approximate \( M + m \approx M \):

\[ 30M = MV_2 \]

Dividing both sides by \( M \):

\[ V_2 = 30 \, \text{m/s} \]

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

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