A body travels a distance x in first two seconds anda distance y in next two seconds. The relation between and y is

Options

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

Explanation

To find the relation between the distances \(x\) and \(y\) traveled by a body in two intervals of time, we can follow these steps:

### Step 1: Understand the motion
The body travels a distance \(x\) in the first 2 seconds and a distance \(y\) in the next 2 seconds. We need to establish a relationship between \(x\) and \(y\).

### Step 2: Use the kinematic equation
The kinematic equation for distance traveled under uniform acceleration is given by:
\[ s = ut + \frac{1}{2} a t^2 \]
Where:
- \(s\) is the distance traveled,
- \(u\) is the initial velocity,
- \(a\) is the acceleration,
- \(t\) is the time.

Since the body starts from rest, the initial velocity \(u = 0\).

### Step 3: Calculate distance \(x\)
For the first 2 seconds:
\[ x = 0 \cdot 2 + \frac{1}{2} a (2^2) = \frac{1}{2} a \cdot 4 = 2a \]

### Step 4: Calculate distance \(y\)
For the next 2 seconds (from \(t = 2\) seconds to \(t = 4\) seconds), the total distance traveled in 4 seconds is:
\[ s = 0 \cdot 4 + \frac{1}{2} a (4^2) = \frac{1}{2} a \cdot 16 = 8a \]
The distance \(y\) traveled in the second interval (from 2 to 4 seconds) can be expressed as:
\[ y = \text{Total distance in 4 seconds} - \text{Distance in first 2 seconds} = 8a - x \]

### Step 5: Substitute \(x\) into the equation for \(y\)
From Step 3, we have \(x = 2a\). Therefore:
\[ y = 8a - 2a = 6a \]

### Step 6: Relate \(x\) and \(y\)
Now we can express \(y\) in terms of \(x\):
- Since \(x = 2a\), we can find \(a\) in terms of \(x\):
\[ a = \frac{x}{2} \]
- Substitute \(a\) into the equation for \(y\):
\[ y = 6a = 6 \cdot \frac{x}{2} = 3x \]

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