A roller coaster car (390 kg) moves from A (5.00 m above the ground) to B (19.0 m above the ground). Two nonconservative forces are present: friction does -2.00 104 J of work on the car, and a chain mechanism does +3.00 104 J of work to help the car up a long climb. What is the change in the car's kinetic energy, ÄKE = KEf − KE0, from A to B?
The change in potential energy is
Δ PE = mgh2-mgh1 = 390•9.8•(19 -5) = 53508J
If the change in car's kinetic energy is ΔKE, then
ΔKE + 3.0•10^4 - 2.0•10^4 = Δ PE
ΔKE = 53508 -10000 = 43508 J
To find the change in the car's kinetic energy from point A to point B, we need to consider the work done and the change in potential energy.
The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. In this case, we have two nonconservative forces, friction which does negative work, and the chain mechanism which does positive work.
First, let's calculate the work done by each force. The work done by friction is -2.00 * 10^4 J (negative because it does negative work), and the work done by the chain mechanism is +3.00 * 10^4 J (positive because it does positive work).
Next, let's calculate the change in potential energy. The change in potential energy is equal to the difference in the potential energy at point B and point A.
The potential energy at point A is equal to mgh, where m is the mass of the car (390 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height above the ground (5.00 m). Therefore, the potential energy at point A is:
PE(A) = mgh = (390 kg)(9.8 m/s^2)(5.00 m) = 19050 J
Similarly, the potential energy at point B can be calculated as:
PE(B) = mgh = (390 kg)(9.8 m/s^2)(19.0 m) = 72954 J
The change in potential energy is then:
ΔPE = PE(B) - PE(A) = 72954 J - 19050 J = 53904 J
Finally, to find the change in kinetic energy, we subtract the work done by friction and the change in potential energy from the work done by the chain mechanism:
ΔKE = Work(chain) + Work(friction) = 3.00 * 10^4 J - 2.00 * 10^4 J = 1.00 * 10^4 J
Therefore, the change in the car's kinetic energy from point A to point B is 1.00 * 10^4 J.