Step 1: Understanding the Concept:
This problem deals with the distances covered in successive time intervals by a body under uniform acceleration starting from rest. The key is to apply the kinematic equation for displacement.
Step 2: Key Formula or Approach:
The equation for displacement $s$ for a body starting from rest ($u=0$) with uniform acceleration $a$ is:
\[ s = \frac{1}{2}at^2 \]
We will apply this formula for different time intervals.
Step 3: Detailed Explanation:
The body starts from rest, so $u=0$. Let the uniform acceleration be 'a'.
Distance covered in the first 2 seconds (t=2s):
This distance is given as 'x'.
\[ x = \frac{1}{2}a(2)^2 = \frac{1}{2}a(4) = 2a \quad \dots(1) \]
Distance covered in the "next" 2 seconds:
This is the distance covered between $t=2$s and $t=4$s. We can find this by calculating the total distance covered in 4 seconds and subtracting the distance covered in the first 2 seconds.
Total distance covered in 4 seconds ($t_{total}=4$s):
\[ s_{total} = \frac{1}{2}a(4)^2 = \frac{1}{2}a(16) = 8a \]
The distance covered in the first 2 seconds is $x=2a$.
The distance covered in the next 2 seconds is 'y':
\[ y = s_{total} - x = 8a - 2a = 6a \quad \dots(2) \]
Relating y and x:
From equation (1), we have $a = x/2$.
Substitute this into equation (2):
\[ y = 6a = 6\left(\frac{x}{2}\right) = 3x \]
So, the relationship is $y=3x$.
Step 4: Final Answer:
The relationship between the distances is $y=3x$. Therefore, option (C) is correct.