Given:
- Distance covered in the first interval (s1) = 24 m
- Distance covered in the second interval (s2) = 64 m
- Time taken for each interval (t) = 4 seconds

To Find:
- Initial velocity of the particle (u)

Formula:
The formula that relates distance, initial velocity, time, and acceleration is:
s = ut + (1/2)at^2

Where:
- s is the distance traveled
- u is the initial velocity
- t is the time taken
- a is the acceleration

Explanation:
We can calculate the acceleration of the particle using the formula:

Acceleration (a) = (Change in velocity)/(Time taken)

Since the time taken for each interval is the same, we can calculate the change in velocity by dividing the change in distance by the time taken.

Change in velocity (v) = (s2 - s1)/t

Substituting the given values:
v = (64 - 24)/4 = 40/4 = 10 m/s

Now, we can calculate the acceleration:
a = v/t = 10/4 = 2.5 m/s^2

Using the formula for distance, we can rearrange it to find the initial velocity:

s = ut + (1/2)at^2

Since the particle starts from rest (u = 0), the equation simplifies to:

s = (1/2)at^2

Substituting the values for the first interval:
24 = (1/2)(2.5)(4)^2
24 = 10(16)
24 = 160

The equation is not satisfied, which means the initial assumption that the particle started from rest is incorrect. The initial velocity (u) must be non-zero.

To find the correct value of the initial velocity, we can use the equation:

s = ut + (1/2)at^2

Substituting the values for the second interval:
64 = u(4) + (1/2)(2.5)(4)^2
64 = 4u + 20
4u = 44
u = 11 m/s

Answer:
The initial velocity of the particle is 11 m/s.

4m/s