Question

The distance from point A to point B is s. In the motion from A to B and back , the speed for the first part of the motion is v₁ and the speed for the return part of the motion is v₂. The avg speed for the entire motion is v. Prove that v = (2v₁v₂) / (v₁ + v₂).

Answer 1

The average speed, v, is given by v = (v₁t₁ + v₂t₂) / t where t is the total time taken for the round trip. The equation can also be expressed as v = (2s) / t, where s is the distance traveled in each direction.

Step-by-step explanation:

The average speed, v, is defined as the total distance traveled divided by the total time taken. So, the average speed is given by:

v = (s + s) / (t₁ + t₂)
where t₁ is the time taken for the first part of the motion from A to B with speed v₁ and t₂ is the time taken for the return part of the motion from B to A with speed v₂.

Since the distance traveled in each direction is the same, s = s. We can rewrite the equation as:

v = (2s) / (t₁ + t₂)
Using the formula for speed (speed = distance / time), we can also rewrite it as:

v = (2v₁t₁ + 2v₂t₂) / (t₁ + t₂)
Since v is the average speed for the entire motion, we can rewrite it as:

v = (v₁t₁ + v₂t₂) / (t₁ + t₂)
Lastly, we can express t₁ + t₂ as the total time taken for the entire motion, which is the round trip time:

t = t₁ + t₂

Therefore, the equation becomes:

v = (v₁t₁ + v₂t₂) / t