Consider the diagram representing a portion of a roller coaster. A 100 kg car is brought up to the top of the hill at a height of 30 meters above the ground at point A. It moves down a frictionless track and comes to a stop as it compresses a spring at point D. What is the total energy for the ride? How much kinetic energy does the car have at point B? C? What speed does the car have at B and C? How much energy is contained in the spring at the end of the ride? If the ride takes 15 seconds to ascend to the peak up the first ramp, how much power is demanded?
We can start the problem by drawing a qualitative bar chart to depict how the energy is transferred throughout the system. At point A, there is no velocity, therefore there is no kinetic energy. There are also no elastic objects attached to the cart, so there is no potential elastic energy. However, the object is located above level ground, so it does have gravitational potential energy. At point B, some of the gravitational energy has transferred to kinetic, but since the object is still above ground, it still has some gravitational potential energy. This changes at point C, where all of the object's energy is kinetic. Finally, when it reaches a stop at point D, it has gravitational potential energy, and the remainder of the total energy is in the spring.
At the beginning of the ride, all of the cart's energy is in the form of gravitational potential energy. Hence, the system's total energy can be calculated by calculating the initial gravitational energy. This can be done with the formula Ug = mgh.
Ug = mgh
Ug = 100 kg * 9.8 m/sec^2 * 30 m
Ug = 29400J
There is a total of 29400 Joules in the system.
Ug = 100 kg * 9.8 m/sec^2 * 30 m
Ug = 29400J
There is a total of 29400 Joules in the system.
At point B, we have to consider the fact that the total energy (calculated above) is split into both Kinetic energy and gravitational potential energy. It is quite easy to calculate the gravitational potential energy using the same formula, from there, we can subtract it from the total energy and the remaining will be kinetic. This holds true because no energy is lost to the surroundings (because the track is frictionless).
Ug = mgh
Ug = 100 kg * 9.8 m/sec^2 * 10 m
Ug = 9800J
Total Energy - 9800J = Kinetic Energy
29400J - 9800J = 19600J of kinetic energy
From here, it is quite easy to find the velocity at point B. Because Kinetic Energy = (1/2) m * v^2, we can plug in the known values for mass and kinetic energy and solve for the velocity.
Kinetic Energy = (1/2) m * v^2
19600J = (1/2) 100 kg (v^2)
v = 19.799 m/sec
The cart travels at a speed of 19.799 m/sec at point B along the roller coaster.
Ug = 100 kg * 9.8 m/sec^2 * 10 m
Ug = 9800J
Total Energy - 9800J = Kinetic Energy
29400J - 9800J = 19600J of kinetic energy
From here, it is quite easy to find the velocity at point B. Because Kinetic Energy = (1/2) m * v^2, we can plug in the known values for mass and kinetic energy and solve for the velocity.
Kinetic Energy = (1/2) m * v^2
19600J = (1/2) 100 kg (v^2)
v = 19.799 m/sec
The cart travels at a speed of 19.799 m/sec at point B along the roller coaster.
At point C, the cart has reached ground level, and thus has no more gravitational potential energy. This means that the total energy of the system has all been converted into kinetic energy. This allots for 29400J of kinetic energy at point C. A similar strategy to the problem above can be used to solve for the speed of the car. We can use the same formula to find the velocity.
Kinetic Energy = (1/2) m * v^2
29400J = (1/2) 100kg (v^2)
v = 24.249 m/sec
The cart travels at a speed of 24.249 m/sec at point C along the roller coaster.
29400J = (1/2) 100kg (v^2)
v = 24.249 m/sec
The cart travels at a speed of 24.249 m/sec at point C along the roller coaster.
Considering point D, we know that there is no kinetic energy because the cart has come to a stop. This means that all of the energy is potential, but it is divided between gravitational potential and elastic potential energy. We can calculate the gravitational potential energy using the formula Ug = mgh. The energy remaining after this is subtracted from the total energy is what is stored in the spring.
Ug = mgh
Ug = 100 kg *9.8 m/sec^2 * 20 m
Ug = 19600J
Total Energy - Ug = Us (elastic potential)
29400J - 19600J = 9800J
There are 9800J of elastic potential energy in the spring.
Ug = 100 kg *9.8 m/sec^2 * 20 m
Ug = 19600J
Total Energy - Ug = Us (elastic potential)
29400J - 19600J = 9800J
There are 9800J of elastic potential energy in the spring.
To calculate the power that the ride lifted the cart with, we must first find the work that was put into the system. At the base of the ride, the cart had 0 energy. At the top of the hill, the cart had 29400J of energy. Since work is the amount of energy put in or taken out of a system, there was 29400J of work that was done. Power is defined as work/time, so all we need to do to calculate the power is divide the work by the time, both of which are given.
Power = 29400J/ 15 seconds
Power = 1960 watts
The ride has a power of 1960 watts in order to raise the cart up 30m in 15 seconds.
Power = 1960 watts
The ride has a power of 1960 watts in order to raise the cart up 30m in 15 seconds.