A bicyclist pedals at an average speed of 20 mph his first lap around a circle. His second lap was at an average speed of 30 mph. What was his average speed for the entire trip?

Question

I can't seem to grasp how to calculate the avg velocity without the actual distance or the time being given. I know that the distance would equal 2D since he's going the same distance at a different rate but I can't figure out how to incorporate it.

Thanks

Answer 1

24 mph is the average speed

you're tempted to say 25, as (30+20)/2 = 25 but he spent more time at 20, so the average will be a little lower than 25.

pick any number for the distance. pick a number that simplifies the calculation, such as 30 miles. maybe that's very unrealistic but any number works.

that means 1 hour at 30 mph and 1 1/2 hours at 20 miles per hour, for a total 2 1/2 hours

total 60 miles at 2 1/2 hours. that's the same speed as 120 miles at 5 hours

120/5 = 24 mph. BINGO. that's the average speed. 24 MPH

just to convince you, try another number for distance, such as 20 miles

that's 2d = 2(20) = 40 miles total distance. 1 hour at 20 mph and 2/3 hour at 30 mph

total time = 1 2/3 = 5/3

divide 40 by 5/3 to get 40/5/3 = 40x3/5 = 8x3 = 24 MPH.

try any number for distance, you'll get exactly 24 MPH

it also doesn't matter whether it's a circle, an ellipse, a square, a 45 sided polygon or any shape that he rode the bicylce around. uphill, downhill, up and down or whatever. assuming he turns corners without changing speed, another unrealistic assumption

more generally,

he goes distance = 2d

he travels time t = 2d/20 1st lap

then t =2d/30 2nd lap

average of 2d/20 + 2d/30 = (2d/20 +2d/30)/2

= (6d/60 +4d/60)/2 = 10d/60/2 = d/12 hours total

divide distance by time to get speed

2d/(d/12) = 2d(12/d) = 24 mph