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A car travels up a hill at a constant speed of 19 km/h and returns down the hill at a constant speed of 50 km/h. Calculate the average speed for the round trip.

Am I supposed to add the numbers, divide by two, then subtract two?

This is how the book explains a similar problem: https://i.stack.imgur.com/wYAAE.png -> 40 + 60 / 2 is 50, then subtract 2 for 48? Is that what's going on?

user416503
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  • Take the Harmonic mean. – SchrodingersCat Feb 19 '17 at 11:34
  • Have you looked at the image that you have given the link of here? It explains clearly how the average speed is calculated. Average speed is ratio of total distance and total time taken to traverse that distance. And they get $$\frac{2}{\frac{100}{2400}} = 48.$$ Use the same to find a solution to your problem. – rookie Feb 19 '17 at 11:41
  • Well, the link I provided failed to explain how the math works. After all, if Ds cancel, the whole thing would just become 0. – user416503 Feb 19 '17 at 11:53

2 Answers2

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Here the $D$ factor in the average speed just cancels out, giving us, $$\text{ average speed } = \frac{2D}{\frac{D}{v_{\text{up}}} + \frac{D}{v_{\text{down}}}} = \frac{2D}{D(v_{\text{up}} + v_{\text{down}})}(v_{\text{up}} v_{\text{down}})= \frac{2(v_{\text{up}} v_{\text{down}})}{v_{\text{up}} + v_{\text{down}}}$$

Substituting the required values, gives us the average speed as $48$ km/hr. Hope it helps.

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For this problem (as opposed to the worked example from the book), the answer is $\frac{(2)(50)(19)}{50+19} = \frac{1900}{69} \approx 27.54 km/h$

For equal distances travelled at two constant speeds each, the average speed is the harmonic mean of the two speeds.

Deepak
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