Question

When riding Full Throttle, the steel roller coasterat Six Flags Magic Mountain in Valencia, California, riders experience the world's tallest loop at 38.75 m (diameter) (127.13 f). Determine the minimum speed in the loop to keep from falling off.

Answer 1

To keep from falling off at the top of a roller coaster loop, we calculate the minimum speed by equating the centripetal force to gravitational force. Given the radius of curvature of 15.0 m and the desired centripetal acceleration of 1.50 g, the minimum speed required is approximately 14.7 m/s.

Step-by-step explanation:

The student's question involves determining the minimum speed at the top of a roller coaster loop to prevent riders from falling off. To answer this, we must understand that gravity provides the centripetal force needed for the roller coaster car to complete the loop. At the minimum speed, the gravitational force is just enough to keep the car in a circular path without requiring additional forces. Using the formula for centripetal acceleration, a = v^2 / r, where v is velocity, r is the radius of curvature (15.0 m in the example), and a is the acceleration (which is 1.50 times the acceleration due to gravity, g).

First, we find the required centripetal acceleration that equals 1.50 g. Since g is approximately 9.8 m/s^2, we have 1.50 x 9.8 m/s^2 = 14.7 m/s^2. Now, we can solve for v by rearranging the centripetal acceleration formula to v = sqrt(a x r), yielding v = sqrt(14.7 m/s^2 x 15.0 m). Upon calculation, we find that the required minimum speed is approximately 14.7 m/s.