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. y = 3x
B. y = 4x
C. y = x
D. y = 2x

Explanation

To solve the problem, we need to analyze the motion of the particle under uniform acceleration. Let's break down the steps:

### Step 1: Understand the motion of the particle
The particle starts from rest, which means its initial velocity \( u = 0 \). The distance traveled under uniform acceleration can be described by the equation:
\[ s = ut + \frac{1}{2} a t^2 \]
Since \( u = 0 \), this simplifies to:
\[ s = \frac{1}{2} a t^2 \]

### Step 2: Calculate the distance traveled in the first 2 seconds
For the first 2 seconds (\( t = 2 \)):
\[ x = \frac{1}{2} a (2^2) = \frac{1}{2} a \cdot 4 = 2a \]

### Step 3: Calculate the distance traveled in the next 2 seconds
Now, we need to find the distance traveled from \( t = 2 \) seconds to \( t = 4 \) seconds. The total distance traveled at \( t = 4 \) seconds is:
\[ s(t=4) = \frac{1}{2} a (4^2) = \frac{1}{2} a \cdot 16 = 8a \]
The distance traveled during the interval from \( t = 2 \) to \( t = 4 \) seconds is:
\[ y = s(t=4) - s(t=2) = 8a - 2a = 6a \]

### Step 4: Relate \( y \) and \( x \)
From the problem, we know that \( y = nx \). We have already calculated:
- \( x = 2a \)
- \( y = 6a \)

Substituting these into the equation \( y = nx \):
\[ 6a = n(2a) \]

### Step 5: Solve for \( n \)
We can simplify this equation by dividing both sides by \( 2a \) (assuming \( a \neq 0 \)):
\[ n = \frac{6a}{2a} = 3 \]

### Conclusion
Thus, the value of \( n \) is:
\[ \boxed{3} \]