For a particle under going rectilinear motion with uniform acceleration, the magnitude of displacement is one third the distance covered in some time interval. The magnitude of final velocity is less than magnitude of initial velocity for this time interval. Then the ratio of initial speed velocity to the final speed for this time interval is :

Options

A. `sqrt(2)`
B. 2
C. `sqrt(3)`
D. 3

Explanation

To solve the problem, we will follow a systematic approach using the equations of motion and the information provided in the question.

### Step 1: Understanding the Problem
We know that:
- The displacement \( s \) is one-third of the total distance covered \( d \).
- The final velocity \( v \) is less than the initial velocity \( u \).

Let’s denote the total distance covered as \( d = 3s \). Therefore, the displacement is:
\[ s = \frac{1}{3} d = \frac{1}{3} (3s) = s \]

### Step 2: Analyzing the Motion
Assuming the particle moves in two segments:
1. The first segment covers a distance of \( 2s \) (where it decelerates).
2. The second segment covers a distance of \( s \) (where it accelerates back).

### Step 3: Using the Equations of Motion
For the first segment (deceleration):
- Initial velocity = \( u \)
- Final velocity = \( 0 \)
- Distance = \( 2s \)

Using the third equation of motion:
\[ v^2 = u^2 + 2as \]
Substituting \( v = 0 \):
\[ 0 = u^2 - 2a(2s) \]
\[ u^2 = 4as \quad \text{(Equation 1)} \]

For the second segment (acceleration):
- Initial velocity = \( 0 \)
- Final velocity = \( v \)
- Distance = \( s \)

Using the same equation:
\[ v^2 = u^2 + 2as \]
Substituting \( u = 0 \):
\[ v^2 = 0 + 2a(s) \]
\[ v^2 = 2as \quad \text{(Equation 2)} \]

### Step 4: Finding the Ratio of Initial and Final Velocities
Now, we have two equations:
1. \( u^2 = 4as \)
2. \( v^2 = 2as \)

To find the ratio \( \frac{u}{v} \), we can divide Equation 1 by Equation 2:
\[ \frac{u^2}{v^2} = \frac{4as}{2as} \]
This simplifies to:
\[ \frac{u^2}{v^2} = 2 \]
Taking the square root:
\[ \frac{u}{v} = \sqrt{2} \]

### Step 5: Conclusion
Thus, the ratio of the initial speed \( u \) to the final speed \( v \) is:
\[ \frac{u}{v} = \sqrt{2} \]