A motorist traveling on a highway at a speed of 70mph exits

A motorist traveling on a highway at a speed of 70mph exits onto an ice-covered exit ramp. Wishing to stop, he applies his brakes until his automobile comes to rest. Knowing that the magnitude of the total acceleration of the automobile cannot exceed 15ft/s2 , determine the minimum time required for the automobile to come to rest and the distance it travels on the exit ramp during that time for the following case. The exit ramp has a constant radius of curvature of 850ft and the driver applies his brakes so that dv/dt varies linearly

Solution

70 mph = 70 x 5280 / 3600 = 102.67 ft/s

centripetal force when he entered the ramp = (102.67)²/ 850 = 12.40 ft/s²

let the tangential decceleration be x

x² + (12.40)² = 15² => x = 8.44 ft/s²

now x is constant

initial velocity =u = 102.67 ft/s final velocity =v = 0

v = u - at => t = u/a => t = 102.67 /8.44 = 12.16 s is the time the person takes to stop

again

distance = ut - 1/2 at² = 624.47 ft