A particle starts moving from rest under uniform acceleration it travels a distance x in the first two seconds and a distance y in the next two seconds. If `y=nx`, then `n=`

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To solve the problem, we need to analyze the motion of the particle under uniform acceleration. We will use the equations of motion to find the distances traveled in the specified time intervals and then relate them to find the value of \( n \).

### Step-by-Step Solution:

1. **Understand the Motion**:
- The particle starts from rest, which means its initial velocity \( u = 0 \).
- It moves with uniform acceleration \( a \).

2. **Distance Traveled in the First 2 Seconds**:
- We can use the equation of motion:
\[ s = ut + \frac{1}{2} a t^2 \]
- For the first 2 seconds (\( t = 2 \) seconds):
\[ x = 0 \cdot 2 + \frac{1}{2} a (2)^2 = \frac{1}{2} a \cdot 4 = 2a \]
- So, we have:
\[ x = 2a \]

3. **Distance Traveled in the Next 2 Seconds**:
- The total time for this part is 4 seconds. The distance traveled in the first 4 seconds is given by:
\[ s = ut + \frac{1}{2} a t^2 \]
- For \( t = 4 \) seconds:
\[ s = 0 \cdot 4 + \frac{1}{2} a (4)^2 = \frac{1}{2} a \cdot 16 = 8a \]
- The distance traveled in the first 4 seconds is \( 8a \), and we know the distance traveled in the first 2 seconds is \( x = 2a \). Thus, the distance traveled in the next 2 seconds, \( y \), is:
\[ y = \text{Total distance in 4 seconds} - \text{Distance in first 2 seconds} = 8a - 2a = 6a \]

4. **Relate \( y \) to \( x \)**:
- We are given that \( y = nx \).
- Substituting the values of \( y \) and \( x \):
\[ 6a = n(2a) \]
- Dividing both sides by \( 2a \) (assuming \( a \neq 0 \)):
\[ n = \frac{6a}{2a} = 3 \]

5. **Final Answer**:
- Thus, the value of \( n \) is:
\[ n = 3 \]