A truck moving on horizontal road towards east with velocity `20ms^(1)` collides elastically with a light ball moving with velocity `25ms^(-1)` along west. The velocity of the ball just after collision

Options

A. `65ms^(-1)` towards east
B. `25ms^(-1)` towards west
C. `65ms^(-1)` towards west
D. `20ms^(-1)` towards east

Explanation

To solve the problem of determining the velocity of the light ball just after the elastic collision with the truck, we can follow these steps:

### Step 1: Define the System
We have a truck moving east with a velocity of \( v_T = 20 \, \text{m/s} \) and a light ball moving west with a velocity of \( v_B = -25 \, \text{m/s} \) (the negative sign indicates the opposite direction).

### Step 2: Use Conservation of Momentum
In an elastic collision, the total momentum before the collision is equal to the total momentum after the collision. We can express this mathematically as:
\[ P_{\text{initial}} = P_{\text{final}} \]
The initial momentum of the system can be expressed as:
\[ P_{\text{initial}} = M v_T + m v_B = M (20) + m (-25) \]
Where \( M \) is the mass of the truck and \( m \) is the mass of the ball.

### Step 3: Write the Final Momentum
After the collision, let \( V_T \) be the velocity of the truck and \( V_B \) be the velocity of the ball. The final momentum can be written as:
\[ P_{\text{final}} = M V_T + m V_B \]

### Step 4: Set Up the Momentum Conservation Equation
Setting the initial momentum equal to the final momentum gives us:
\[ M (20) - 25m = M V_T + m V_B \]

### Step 5: Use the Coefficient of Restitution
For elastic collisions, the coefficient of restitution \( e = 1 \). This gives us the relationship:
\[ V_B - V_T = - (v_B - v_T) \]
Substituting the values:
\[ V_B - V_T = -(-25 - 20) \]
\[ V_B - V_T = 45 \]
This can be rearranged to:
\[ V_B = V_T + 45 \]

### Step 6: Substitute for \( V_T \)
Since the mass of the ball is negligible compared to the mass of the truck, we can assume that the truck's velocity \( V_T \) does not change significantly. Thus, we can approximate \( V_T \) to be \( 20 \, \text{m/s} \):
\[ V_B = 20 + 45 = 65 \, \text{m/s} \]

### Step 7: Conclusion
The velocity of the ball just after the collision is:
\[ V_B = 65 \, \text{m/s} \text{ towards East} \]