A truck and a car having equal kinetic energies are stopped by applying equal retarding force. If `S_(1)` and `S_(2)` are the distances covered by truck and car respectively, before stopping, then

Options

A. `S_(1)=S_(2)`
B. `S_(1) gt S_(2)`
C. `S_(1) lt S_(2)`
D. Cannot say

Explanation

To solve the problem, we need to establish the relationship between the distances \( S_1 \) and \( S_2 \) covered by the truck and the car, respectively, before stopping, given that they have equal kinetic energies and are stopped by equal retarding forces.

### Step-by-Step Solution:

1. **Understanding Kinetic Energy**:
The kinetic energy (KE) of an object is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
where \( m \) is the mass and \( v \) is the velocity of the object.

2. **Setting Up the Problem**:
Let:
- \( M_1 \) = mass of the truck
- \( U_1 \) = initial velocity of the truck
- \( M_2 \) = mass of the car
- \( U_2 \) = initial velocity of the car

Since both vehicles have equal kinetic energies:
\[ \frac{1}{2} M_1 U_1^2 = \frac{1}{2} M_2 U_2^2 \]
This simplifies to:
\[ M_1 U_1^2 = M_2 U_2^2 \quad \text{(1)} \]

3. **Using the Equation of Motion**:
The equation of motion that relates initial velocity, final velocity, acceleration, and distance is:
\[ v^2 = u^2 + 2as \]
Here, the final velocity \( v = 0 \) (since both vehicles stop), so we can rewrite it for both vehicles.

For the truck:
\[ 0 = U_1^2 + 2(-a_1) S_1 \implies S_1 = \frac{U_1^2}{2a_1} \quad \text{(2)} \]
For the car:
\[ 0 = U_2^2 + 2(-a_2) S_2 \implies S_2 = \frac{U_2^2}{2a_2} \quad \text{(3)} \]

4. **Relating the Retarding Forces**:
Since both vehicles are stopped by equal retarding forces, we can express the retardation as:
\[ a_1 = \frac{F}{M_1} \quad \text{and} \quad a_2 = \frac{F}{M_2} \]
Substituting these into equations (2) and (3):
\[ S_1 = \frac{U_1^2}{2 \left( \frac{F}{M_1} \right)} = \frac{M_1 U_1^2}{2F} \quad \text{(4)} \]
\[ S_2 = \frac{U_2^2}{2 \left( \frac{F}{M_2} \right)} = \frac{M_2 U_2^2}{2F} \quad \text{(5)} \]

5. **Finding the Ratio \( \frac{S_1}{S_2} \)**:
Now we can find the ratio of the distances:
\[ \frac{S_1}{S_2} = \frac{\frac{M_1 U_1^2}{2F}}{\frac{M_2 U_2^2}{2F}} = \frac{M_1 U_1^2}{M_2 U_2^2} \]
From equation (1), we have \( M_1 U_1^2 = M_2 U_2^2 \). Therefore:
\[ \frac{S_1}{S_2} = \frac{M_1 U_1^2}{M_2 U_2^2} = 1 \]

6. **Conclusion**:
Thus, we find that:
\[ S_1 = S_2 \]