Question

A rollercoaster loop has a radius of 22.7 m. what is the minimum speed the coaster must have at the top of the loop to not fall off the track? (unit = m/s)

Answer 1

v = 14.92 m/s

Step-by-step explanation:

First, make a free body diagram and see the forces in the y-direction.

• ∑F_y = F_N - F_g

Use Newton's 2nd Law F = ma to replace ∑F_y with m * a_y.

• m * a_y = F_N - F_g

The acceleration in the y-direction is the centripetal acceleration, a_c = v^2/r.

• m * v^2/r = F_N - mg

The normal force is 0 because this is where the rollercoaster is not falling off the track yet not touching the track.

• m * v^2/r = - mg

The masses cancel out.

• v^2/r = -g

Solve for v to find the speed of the rollercoaster at the top of the loop.

• v^2 = -(-9.81) * r
• v = √(9.81 * 22.7)
• v = 14.9227

The minimum speed the coaster must have at the top of the loop to not fall off the track is 14.92 m/s.