Final answer:
The speed of the roller coaster car at the top of the loop is calculated based on the conditions that its apparent weight equals its actual weight. Using circular motion equations and the provided loop diameter, the speed is found to be approximately 14 meters per second.
Step-by-step explanation:
The student is asking about the speed of a roller coaster car at the top of a loop when its apparent weight is equal to its actual weight. To find this speed, we can use the principles of circular motion and Newton's second law. When a roller coaster car is at the top of a loop, the forces acting on it are gravity (pulling it down) and the normal force from the loop (also acting downward if the car's apparent weight is equal to its actual weight). The sum of these forces provides the centripetal force required for the car to travel in a circular path.
At the top of the loop, the equation for centripetal force (Fc) is Fc = mv^2/r, where m is mass, v is velocity, and r is the radius of the loop. Since the car's apparent weight is equal to its actual weight at this point, we can set the centripetal force equal to the weight of the car, mg, where g is the acceleration due to gravity. This gives us mg = mv^2/r. Simplifying, we get v = sqrt(gr). Plugging in the values, where g = 9.81 m/s^2 and r = 40 m / 2 (the radius is half the diameter), we find the speed of the roller coaster car at the top of the loop.
v = sqrt(9.81 m/s^2 * (40 m / 2))
v = sqrt(9.81 m/s^2 * 20 m)
v = sqrt(196.2)
v = 14.007 m/s
Thus, the roller coaster car's speed at the top of the loop is approximately 14 meters per second.