Speed, distance and time

Driving 60 km out at 60 km/h and back at 40 km/h averages 48 km/h, not 50.

Out at 60 km/h, back at 40. What is the average?

Answer this before reading further. You drive 60 km to a town at 60 km/h and drive the same road back at 40 km/h. What was your average speed for the whole trip? If you wrote 50, keep that number nearby. It is the obvious answer, and it is wrong. By the end of the page the real answer, 48, should look obvious too.

150 km in 2 hours is 75 km in one

A train covers 150 km in 2 hours. Its average speed is the distance per one hour: 150 ÷ 2 = 75 km/h. Speed is the most familiar unit rate there is, and OpenStax uses it as the example when it introduces unit rates: 68 miles per hour means 68 miles in 1 hour. Any unit of distance over any unit of time works: metres per second, kilometres per hour, miles per hour.

d = rt, and the two ways to rearrange it

OpenStax writes the formula for an object moving at a constant speed as d = rt: distance equals rate times time. Solve it for the other letters and you get r = d/t and t = d/r. At 70 km/h for 3 hours you cover 70 × 3 = 210 km. To cover 210 km at 70 km/h takes 210 ÷ 70 = 3 hours. And 210 km in 3 hours is 210 ÷ 3 = 70 km/h. The three forms are the same fact seen from different sides.

45 minutes is 0.75 hours, not 0.45

Minutes and hours are easy to mix up. A cyclist rides 30 km in 45 minutes. 30 ÷ 45 gives 0.667 km per minute, which is correct but is not km/h. Change the time to hours first: 45 minutes is 45/60 = 0.75 hours, and 30 ÷ 0.75 = 40 km/h. Going the other way, 1.2 hours is not 1 hour 20 minutes. It is 1 hour and 0.2 × 60 = 12 minutes. A made-up ride to school, 3 km in 15 minutes, works the same way: 15 minutes is 0.25 hours, so 3 ÷ 0.25 = 12 km/h.

Runners often use pace, the rate turned upside down: minutes per kilometre. A pace of 5 minutes per km means 60 ÷ 5 = 12 km in an hour, so 12 km/h. For speeds in metres per second, multiply by 3,600 seconds in an hour and divide by 1,000 metres in a kilometre, which is the same as multiplying by 3.6: 10 m/s is 36 km/h.

Average speed is not the average of the speeds

You drive 60 km to a town at 60 km/h and come back the same 60 km at 40 km/h. The average of 60 and 40 is 50, but that is not your average speed. Find total distance over total time. Going takes 60 ÷ 60 = 1 hour; returning takes 60 ÷ 40 = 1.5 hours. Total: 120 km in 2.5 hours, so 120 ÷ 2.5 = 48 km/h. The slow leg lasts longer, so it pulls the average down more.

60 mph is about 97 km/h

NIST Handbook 44 lists 1 mile as exactly 1.609344 kilometres. A speed of 60 miles per hour is 60 × 1.609344 = 96.56 km/h, about 97. For a rough figure in your head, multiply miles by 1.6, or multiply kilometres by 5/8 to get miles: 80 km/h is about 50 mph.