If a material point moved with the speed v1 for the first half of the way, and v2 for the second

If a material point moved with the speed v1 for the first half of the way, and v2 for the second, then the average speed of the point on all the way is equal to?

Given:

v1 – speed of a material point in the first half of the way;

v2 is the speed of a material point in the second half of the way.

It is required to determine Vav – the average speed of the point along the entire path.

Let S be the entire path traversed by a material point.

Then, the first half of the way the material point passed for:

t1 = S / (2 * v1), and the second half of the journey in time t2 = S / (2 * v2).

The total time spent by the point for the entire journey will be equal to:

t = t1 + t2 = S / (2 * v1) + S / (2 * v2) = S * (v2 + v1) / (2 * v1 * v2).

Then the average speed will be equal to:

Vav = S / t = S / (S * (v2 + v1) / (2 * v1 * v2)) = 2 * v1 * v2 / (v1 + v2).

Answer: the average speed will be 2 * v1 * v2 / (v1 + v2).