length of the total distance the particle traveled

Question

A particle is moving on the perimeter of a circle with radius 𝑟=8 with angular speed of 𝜋9 radians per second. After completing 3 full rotations, the particle traveled for 2 more seconds and stopped. What is the length of the total distance the particle traveled?

Answer 1

The length of 1 rotation is the circumference of a circle, which is 2πr = 2π × 8 = 16π.

So the length of 3 rotations is 3 ×16π = 48π.

In order to calculate the length it travels in 2 seconds, we have to calculate the linear speed v, which is the radius r × the angular speed ω:

v = rω = 8 × 9π = 72π.

So: distance = linear speed × time = 72π × 2 seconds = 144π

Next we add the two distances: 48π + 144π = 192π