Answer to Question #117628 in Physics for Emmanuel

Question

A point traversed half the distance with a velocityV0 . The remaining part of the distance was
covered with velocity V1 for half the time, and with velocity V2 for the other half of the time.
Find the mean velocity of the point averaged over the whole time of motion

Answer 1

Split the motion of the particles into several pieces characterized by corresponding velocity. Assume that the total distance is D, the time to cover the rest of the distance is 2t.

Part 1. A point traversed half the distance with a velocity V0:

$$\frac{D}2=t_1v_0,\\ \space\\ t_1=\frac{D}{2v_0}.$$

Parts 2 and 3: The remaining part of the distance was covered with velocity V1 for half the time, and with velocity V2 for the other half of the time:

$$\frac{D}{2}=tv_1+tv_2=t(v_1+v_2),\\ \space\\ t=\frac{D}{2(v_1+v_2)}.$$

The total distance over total time is

$$v=\frac{D}{T}=\frac{D}{t_1+2t}=\frac{D}{\frac{D}{2v_0}+2\frac{D}{2(v_1+v_2)}}=\frac{2v_0(v_1+v_2)}{2v_0+v_1+v_2}.$$