A carousel rider sits 4.0 m from the center. Each revolution, she comes back to the same spot, having gone the whole way around her circle.
Why does the speed formula work? Where do its distance and its time come from?
Start from the ride itself. The rider sits 4.0 m from the center, and in one revolution she goes around the circle exactly once.
Going around once covers the distance around the circle: the circumference, 2πr. For this rider, that is 2 × π × 4.0 = 25 m.
The time for that one revolution has a name you know: the period, T.
Speed is distance traveled divided by the time taken.
Take the distance of one revolution over the time of one revolution: 2πr over T.
The 2πr in the formula is a distance: the trip around the circle. The T is the time that trip takes.
The formula needs a steady speed. Then any one revolution tells you the speed for the whole motion.
A model boat circles a fountain at a steady speed on a circle of radius 1.5 m. Enter the distance it travels in one period, in m, to the nearest 0.1 m. Answers within 0.1 m are accepted.
A dirt bike rounds a circular loop at a steady speed. Its speed equals 2πr divided by T. Why does the 2πr sit on top?
A go-kart rounds a circular track of radius 10 m at a steady speed. Explain why its speed equals 2π × 10 m divided by its lap time.
In one lap the go-kart goes around the circle exactly once.
One lap covers the circumference: 2π × 10 m.
The lap time is the period: the time for that one trip.
Speed is distance divided by time, so the speed is the circumference divided by the lap time.
The speed is the one-lap distance, 2π × 10 m, over the one-lap time.
Henrik reads v = 2πr/T and asks what the 2πr stands for. Answer him.
2πr is the circumference: the distance around the circle of radius r.
In one period the object travels exactly that distance.
So the formula is the one-revolution distance over the one-revolution time.
2πr is the distance covered in one period: one full circumference.
A go-kart's circular course is rebuilt with twice the radius, and the go-kart still takes the same time per lap, moving steadily. What happens to the distance it covers in one lap?
A wind turbine blade tip circles at a steady speed. Which statement explains why its speed equals 2πr/T?
A BMX rider circles a flat circular pad at a steady speed, 0.75 m from its center. Enter the distance she covers in one period, in m, to the nearest 0.1 m. Answers within 0.1 m are accepted.
A remote-control truck drives a circular course of radius 2.0 m at a steady speed and completes 5 revolutions. Enter the total distance it covers, in m, to the nearest 1 m. Answers within 1 m are accepted.