Multiple choice

A person travels up hill at an average speed of (u - v) km/hr and comes down the hill back at an average speed of (u + v) km/hr. His average speed for the whole journey is ___________________.

  1. u km/hr
  2. $\displaystyle\frac{u^2-v^2}{u}\:km/hr$
  3. $\sqrt{u^2-v^2}\:km/hr$
  4. $\displaystyle\frac{u}{u^2-v^2}\:km/hr$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Average speed = Total distance / Total time. Let distance of hill be d. Time up = d/(u-v), time down = d/(u+v). Total time = d/(u-v) + d/(u+v) = d(u+v+u-v)/(u^2-v^2) = 2du/(u^2-v^2). Average speed = 2d / (2du/(u^2-v^2)) = (u^2-v^2)/u.

AI explanation

For a journey of equal distances covered at two different speeds, the average speed is found using the harmonic mean formula: 2ab divided by the sum of a and b. Substituting the uphill speed of the quantity u minus v and the downhill speed of the quantity u plus v gives a numerator of 2 times the quantity u minus v times the quantity u plus v. The denominator simplifies to u minus v plus u plus v, which is 2u. Canceling the 2 and expanding the numerator to u squared minus v squared gives an average speed of u squared minus v squared all over u kilometers per hour.

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