A particle starts moving from rest with uniform acceleration. It travels a distance x in the first 2 sec and a distance y in the next 2 sec. Then

Options

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

Answer

To solve the problem, we need to find the relationship between the distances \( x \) and \( y \) traveled by a particle moving with uniform acceleration. Let's go through the solution step by step.

### Step 1: Understand the given information
- The particle starts from rest, which means the initial velocity \( u = 0 \).
- The particle has a uniform acceleration \( a \).
- The distance traveled in the first 2 seconds is \( x \).
- The distance traveled in the next 2 seconds (from 2 seconds to 4 seconds) is \( y \).

### Step 2: Calculate the distance \( x \)
Using the equation of motion for distance:
\[ s = ut + \frac{1}{2} a t^2 \]
For the first 2 seconds:
- \( u = 0 \)
- \( t = 2 \) seconds

Substituting these values into the equation:
\[ x = 0 \cdot 2 + \frac{1}{2} a (2^2) \]
\[ x = \frac{1}{2} a \cdot 4 = 2a \]

### Step 3: Calculate the distance \( y \)
The distance \( y \) is the distance traveled from \( t = 2 \) seconds to \( t = 4 \) seconds. We can calculate this using the positions at \( t = 4 \) seconds and \( t = 2 \) seconds:
\[ y = s_4 - s_2 \]
First, calculate \( s_4 \):
\[ s_4 = 0 \cdot 4 + \frac{1}{2} a (4^2) = \frac{1}{2} a \cdot 16 = 8a \]
Now, we already calculated \( s_2 \) as \( x = 2a \).

So, substituting these values:
\[ y = s_4 - s_2 = 8a - 2a = 6a \]

### Step 4: Relate \( x \) and \( y \)
Now we have:
- \( x = 2a \)
- \( y = 6a \)

To find the relationship between \( x \) and \( y \):
\[ y = 6a = 3 \cdot 2a = 3x \]

### Final Result
Thus, the relationship between the distances \( x \) and \( y \) is:
\[ y = 3x \]