Question

Participant 1 rides a fourth roller coaster as shown below. What is the minimum ramp height H if the ride at the top of the loop maintains the minimum speed needed to stay on the track throughout the loop?

A. 10 m
B. 20 m
C. 30 m
D. 35 m

Answer 1

Participant 1 rides a fourth roller coaster as shown below. What is the minimum ramp height H if the ride at the top of the loop maintains the minimum speed needed to stay on the track throughout the loop.

A. 10 meters

B. 20 meters

C. 30 meters

D. 35 meters

Step-by-step explanation:

To determine the minimum speed needed at the top of the loop for the roller coaster to maintain contact with the track, we need to consider that the centripetal force required to move in a circle is provided entirely by the weight of the roller coaster at the minimum speed. At this speed, the normal force will be zero. We use the equation Fc = m * v2 / r, where Fc is the centripetal force, m is the mass, v is the speed, and r is the radius.

The centripetal force here is equal to the gravitational force, which is m * g, so we have m * g = m * v2 / r. Simplifying by cancelling out the mass, we get the minimum speed needed at the top of the loop as v = sqrt(g * r). Plugging in g = 9.8 m/s2 and r = 15.0 m, we can calculate the speed.

However, the student's question lacked some specific details like the total height of the loop and whether we are considering the potential energy at that height or just the kinetic energy at the top of the loop for the determination of minimum ramp height. Nevertheless, the minimum ramp height should be slightly higher than the loop's height to account for the conservation of energy if we assume that the coaster starts from rest. Without additional details, it is hard to provide a definitive answer.

Answer 2

The minimum ramp height H for a roller coaster to maintain minimum speed at the top of a loop can be calculated using the principles of conservation of energy and the requirements for centripetal force, factoring in gravity and initial potential energy.

Step-by-step explanation:

Calculating Minimum Ramp Height for a Roller Coaster

To determine the minimum ramp height H needed for a roller coaster to maintain minimum speed at the top of a loop, we can utilize principles from physics, particularly conservation of energy and the requirements for centripetal force. The key concept is that at the top of the loop, the roller coaster must have enough speed to supply the centripetal force necessary to keep the coaster on the track without falling. This speed (v) can be found by setting the centripetal force equal to the force of gravity at the top of the loop (mv2/r = mg).

Additionally, we consider that the potential energy at the start (mgh) will convert into kinetic energy (1/2mv2) as the coaster descends. Therefore, by using these relationships, we can solve for the initial height. If the loop's radius r is known, and considering that gravitational acceleration g is 9.8 m/s2, we can start by finding the minimum speed at the top of the loop using v2 = rg. Then, we set the potential energy equal to kinetic energy (mgh = 1/2mv2) to solve for the initial height H.