A person travels first half time with speed u, and next half time with speed v. Find the avg speed

Answer 1

I'll assume that the two time increments are indeed 1/2 the total time,

The average speed is the total distance traveled divided by the total elapsed time. Let d = total distance traveled; let d₁ = distance traveled in the first half time; let d₂ = distance traveled in the second half time; let t = the total elapsed time (so each half is 1/2t).

We know that distance = speed multiplied by time. We also know that speed = distance/time

d = d₁ + d₂

d₁ = (u)(1/2t)

d₂ = (v)(1/2t)

d = (u)(1/2t) + (v)(1/2t) = t(1/2u + 1/2v) = t(u + v)/2

avg speed = d/t = t(u + v)/2/t = (u + v)/2

This seems sort of dumb going through this whole thing when you could have just taken the 2 speeds and averaged them to get the answer. However, this ONLY works because the two time increments were the same. If the person traveled 3 hours at speed u and 2 hours at speed v, it would have been different.