A body of mass m is travelling with a velocity vu. When a constant retarding force F is applied, it comes to rest after travelling a distance `s_(1)`. If the initial velocity is 2u with the same force F, the distance travelled before it comes to rest is `s_(2)`. then,

Options

A. `s_(2)=4s_(1)`
B. `2s_(2)=2s_(1)`
C. `s_(2)=(s_(1))/(2)`
D. `s_(2)=s_(1)`

Explanation

To solve the problem, we need to analyze the motion of the body under the influence of a constant retarding force. We will use the kinematic equations of motion to find the relationship between the distances \( s_1 \) and \( s_2 \).

### Step-by-Step Solution:

1. **Identify the Variables:**
- Mass of the body: \( m \)
- Initial velocity in the first case: \( v_1 = u \)
- Initial velocity in the second case: \( v_2 = 2u \)
- Retarding force: \( F \)
- Distance travelled in the first case: \( s_1 \)
- Distance travelled in the second case: \( s_2 \)
- Final velocity in both cases: \( v = 0 \)

2. **Apply the Kinematic Equation for the First Case:**
We can use the equation:
\[ v^2 = u^2 + 2as \]
For the first case:
- Final velocity \( v = 0 \)
- Initial velocity \( u = u \)
- Distance \( s = s_1 \)
- Acceleration \( a = -\frac{F}{m} \) (negative because it's a retarding force)

Plugging these into the equation:
\[ 0 = u^2 + 2\left(-\frac{F}{m}\right)s_1 \]
Rearranging gives:
\[ u^2 = \frac{2F}{m}s_1 \quad \text{(1)} \]

3. **Apply the Kinematic Equation for the Second Case:**
For the second case:
- Final velocity \( v = 0 \)
- Initial velocity \( u = 2u \)
- Distance \( s = s_2 \)
- Acceleration \( a = -\frac{F}{m} \)

Using the same kinematic equation:
\[ 0 = (2u)^2 + 2\left(-\frac{F}{m}\right)s_2 \]
Rearranging gives:
\[ 4u^2 = \frac{2F}{m}s_2 \quad \text{(2)} \]

4. **Relate the Two Equations:**
From equation (1):
\[ s_1 = \frac{mu^2}{2F} \quad \text{(3)} \]
From equation (2):
\[ s_2 = \frac{4mu^2}{2F} = \frac{2mu^2}{F} \quad \text{(4)} \]

5. **Find the Relationship Between \( s_1 \) and \( s_2 \):**
Now, substituting equation (3) into equation (4):
\[ s_2 = 4s_1 \]

### Final Result:
Thus, the relationship between \( s_1 \) and \( s_2 \) is:
\[ s_2 = 4s_1 \]