Introduction:
When a particle starts from rest with constant acceleration, its motion can be described by the equations of motion. In this case, we are interested in finding the ratio of space average velocity to the time average velocity.

Understanding the concept:
To understand the concept, let's define the terms mentioned in the question:
- Space average velocity: It is the average velocity of the particle over a given displacement or distance.
- Time average velocity: It is the average velocity of the particle over a given time interval.

Deriving the equations:
To find the ratio of space average velocity to time average velocity, we can use the equations of motion. Let's assume the initial velocity of the particle is u, the acceleration is a, the displacement is s, and the time taken is t.

Using the equations of motion, we have:
- v = u + at (equation 1)
- s = ut + (1/2)at^2 (equation 2)

Solving for space average velocity:
The space average velocity can be calculated by dividing the total displacement by the total time taken:
- Space average velocity = Total displacement / Total time

From equation 2, we can express the total displacement as:
- Total displacement = s

From equation 1, we can express the total time taken as:
- Total time = t

Therefore, the space average velocity is simply:
- Space average velocity = s / t

Solving for time average velocity:
The time average velocity can be calculated by dividing the total displacement by the total time taken:
- Time average velocity = Total displacement / Total time

From equation 2, we can express the total displacement as:
- Total displacement = s

From equation 1, we can express the total time taken as:
- Total time = t

Therefore, the time average velocity is also:
- Time average velocity = s / t

Ratio of space average velocity to time average velocity:
Since the space average velocity and time average velocity are both equal to s/t, the ratio of space average velocity to time average velocity is:
- Space average velocity / Time average velocity = (s / t) / (s / t) = 1

Conclusion:
The ratio of space average velocity to time average velocity is always equal to 1 when a particle starts from rest with constant acceleration. This means that the space average velocity and time average velocity are equal in this scenario.

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