Once round the merry-go-round
Average speed and average velocity in one revolution.
Back where you started
A child rides a merry-go-round for one full turn and gets off at the same place where she got on.
Do you think her displacement for the full turn is large or small?
What this lesson covers
The idea
In one revolution of a circular path of radius R, the distance travelled is 2πR and the displacement is zero, so the average speed is vav = 2πR/T, T being the time per revolution, and the average velocity is zero.
Back where you started
A child rides a merry-go-round for one full turn and gets off at the same place where she got on.
Do you think her displacement for the full turn is large or small?
- It is zero, since she is back at the start
- It is as long as the whole circle
- It is equal to the radius
One full turn
Set the radius and the time for one turn. Press Play.
Distance round, displacement zero
The path of one revolution was the whole circle, so the distance travelled was its circumference, 2πR. But she ended where she began, so her displacement was zero. The average speed uses the distance and the average velocity uses the displacement.
In one revolution of a circular path of radius R the distance travelled is 2πR and the displacement is zero. So the average speed is vav = 2πR ÷ T, T being the time for one revolution, and the average velocity is zero.
Notes
In one revolution: distance = 2πR, displacement = 0. Average speed = 2πR ÷ T; average velocity = 0.
Check yourself
A child on a merry-go-round of radius 2 m takes 4 s for one revolution. How far does she travel in one revolution? (Use π = 3.14.)
Answer: 12.56 m
2πR = 2 × 3.14 × 2 m = 12.56 m.
For the same child (R = 2 m, T = 4 s), what is the average speed? (Use π = 3.14.)
Answer: 3.14 m s⁻¹
2πR ÷ T = 12.56 m ÷ 4 s = 3.14 m s⁻¹.
What is the displacement after one full revolution?
What is the average velocity over one full revolution?
- 2πR, the length of the circle. That is the distance travelled, not the displacement.
- Zero, because she comes back to the starting point — correct. Yes!
- Equal to the radius R. The displacement is the net change in position. She is back at the start, so it is not the radius.
- 2πR ÷ T, the same as the average speed. That is the average speed, which uses the distance. The average velocity uses the displacement, which is zero.
- R ÷ T. The displacement is zero, so the average velocity is zero, not R ÷ T.
- Zero — correct. Yes!