A Formula One race car is travelling at 89.0 metres per second along a straight track enters a turn on the race track with radius of curvature of 200.0 metres. What centripetal acceleration must a car have to stay on track?
We’re told in this problem statement that the car is moving at 89.0 metres per second; we will call that 𝑣. With the car moving at this speed, it encounters a turn with a curvature radius of 200.0 metres; we will call that 𝑟. We want to find the centripetal acceleration the car must have to stay on track; we will call that 𝑎 sub 𝑐.
Let’s start by drawing an overhead diagram of the car on the turn. Looking down on the car from above, it’s travelling along the track with a linear speed we’ve called 𝑣 of 89.0 metres per second. It enters a turn of radius 200.0 metres. While the car is in that turn, it experiences a centre seeking or centripetal acceleration; that’s the quantity we want to solve for. And to do that, let’s recall the definition for centripetal acceleration.
When an object has centripetal acceleration, that means it’s moving in a circular arc. The arc being a segment of a circle has a radius 𝑟. The centripetal acceleration of an object travelling in a circular arc is equal to the speed of that object squared divided by the radius of the circle in which it travels. So in our scenario, 𝑎 sub 𝑐 being 𝑣 squared over 𝑟 is equal to 89.0 metres per second squared divided by 200.0 metres.
When we enter these values on our calculator, we find a centripetal acceleration of 39.6 metres per second squared. If the car’s centripetal acceleration was less, it would go off track to the outside of the track. If the car’s centripetal acceleration were greater, it would go off track to the inside of the track. So this value we’ve solved for is the exact centripetal acceleration needed for the car to stay on a curve of this radius moving at this speed.