Prove that in rectilinear uniformly accelerated motion, the average speed of the body is equal

Prove that in rectilinear uniformly accelerated motion, the average speed of the body is equal to the half-sum of the initial and final speeds.

Given:

The body moves uniformly and rectilinearly;

v0 — initial velocity of the body, m / s;

v1 is the final velocity of the body, m / s;

a – acceleration of the body, m / s ^ 2.

It is required to prove that Vav = (v0 + v1) / 2 (the average velocity of the body is equal to the half-sum of the initial and final velocities of the body).

Let the body be in motion during the time interval t.

Then, the average speed of the body is equal to:

Vav = S / t, where S is the complete path traversed by the body during time t.

Vav = (v0 * t + a * t ^ 2/2) / t = v0 + a * t / 2 = (2 * v0 + a * t) / 2.

Since v0 + a * t = v1, substituting this value, we get:

Vav = (2 * v0 + a * t) / 2 = (v0 + v0 + a * t) / 2 = (v0 + v1) / 2.

Q.E.D.