The radius of curvature of a loop-the-loop for a roller coaster is 14.0 m. At the top of the loop, the force that the seat exerts on a passenger of mass m is 0.35mg. Find the speed of the roller coaster at the top of the loop. (Enter your answers from smallest to largest.)

At the top of the coaster, there is a net force on the passenger of 0.35mg. The centifugal force is of course mV^2/r. The force exerted by seat on passenger can be either a downward or an upward net force. We solve each case separately

Doing a force balance on the chair in the Y direction at the top of the loop

mV^2/r = .35mg + mg (downward seat force on passenger)

mV^2/r = -0.35mg + mg (upward seat on passenger)

Solve separately to get two different values for velocity. In each case the values of m will cancel out.