Introduction:
When a particle starts from rest and undergoes constant acceleration, there are two types of average velocities that can be considered: space average velocity and time average velocity. The ratio of these two velocities can be determined by analyzing the motion of the particle.

Explanation:
To understand the ratio of space average velocity to time average velocity, let's break down the concept and analyze it step by step:

1. Space Average Velocity:
Space average velocity is the average velocity over a certain distance or displacement. It can be calculated by dividing the total displacement by the total time taken.

2. Time Average Velocity:
Time average velocity is the average velocity over a certain time interval. It can be calculated by dividing the total displacement by the total time taken.

3. Constant Acceleration:
When a particle undergoes constant acceleration, its velocity increases or decreases by the same amount in equal time intervals. It means that the change in velocity is constant throughout the motion.

4. Relationship between Velocity, Acceleration, and Time:
In the case of constant acceleration, the relationship between velocity, acceleration, and time can be described by the equation:

v = u + at

where v is the final velocity, u is the initial velocity (in this case, zero as the particle starts from rest), a is the acceleration, and t is the time.

5. Deriving the Ratio:
To determine the ratio of space average velocity to time average velocity, we need to consider the equations of motion.

Let's consider the motion of the particle over a certain time interval t1 to t2, with a constant acceleration a. The initial velocity u is zero, and the final velocity v can be calculated using the equation:

v = u + at

The displacement during this time interval can be calculated using the equation of motion:

s = ut + 0.5at^2

The space average velocity can be calculated by dividing the displacement by the total time taken:

Vspace = s / (t2 - t1)

The time average velocity can be calculated by dividing the displacement by the total time taken:

Vtime = s / (t2 - t1)

Therefore, the ratio of space average velocity to time average velocity is:

Vspace / Vtime = (s / (t2 - t1)) / (s / (t2 - t1))

Simplifying this expression, we get:

Vspace / Vtime = 1

Conclusion:
In the case of a particle starting from rest with constant acceleration, the ratio of space average velocity to time average velocity is equal to 1. This means that both average velocities are the same, indicating that the particle covers equal distances in equal time intervals.

Average of a physical quantity with respect to any other physical quantity requires the interval specified for the later one. once that is defined they question will make sense.