A roller coaster (491 kg) moves from A (1.09 m above the ground) to B (26.3 m above the ground). Two non-conservative forces are present: friction does -2.39 x 104 J of work on the car, and a chain mechanism does +5.18 x 104 J of work to help the car up a long climb. What is the change in the car's kinetic energy KEf - KEo from A to B?
To find the change in the car's kinetic energy, we need to calculate both the initial and final kinetic energy and then find their difference.
The initial kinetic energy (KEo) is the energy possessed by the car at point A, while the final kinetic energy (KEf) is the energy possessed by the car at point B.
The formula for kinetic energy is:
KE = (1/2) * m * v^2
Where:
KE is the kinetic energy
m is the mass of the object
v is the velocity of the object
First, let's find the initial kinetic energy:
At point A, the car is at rest, so its initial velocity (vo) is 0 m/s.
Using the formula, the initial kinetic energy (KEo) is:
KEo = (1/2) * m * vo^2
= (1/2) * 491 kg * (0 m/s)^2
= 0 J (Joules)
Next, let's find the final kinetic energy:
At point B, the car has gained potential energy due to the climb.
The potential energy gained is equal to the work done on the car by the chain mechanism.
Therefore, the difference in potential energy is +5.18 x 10^4 J.
The work done is equal to the change in potential energy:
Work = Change in Potential Energy
5.18 x 10^4 J = m * g * h
(Where h is the change in height)
Rearranging the equation to solve for the change in height,
h = Work / (m * g)
= 5.18 x 10^4 J / (491 kg * 9.8 m/s^2)
≈ 10.62 m
The change in height (h) is approximately equal to 10.62 meters.
Now, let's calculate the final velocity (vf) at point B.
Using the conservation of energy principle, the change in potential energy is equal to the change in kinetic energy:
Change in Potential Energy = Change in Kinetic Energy
Change in Potential Energy = 5.18 x 10^4 J
Change in Kinetic Energy = KEf - KEo = KEf - 0 J
Therefore, KEf = 5.18 x 10^4 J
Finally, let's calculate the final velocity (vf):
Using the formula for kinetic energy:
KEf = (1/2) * m * vf^2
5.18 x 10^4 J = (1/2) * 491 kg * vf^2
Rearranging the equation to solve for vf,
vf = √(2 * KEf / m)
= √(2 * 5.18 x 10^4 J / 491 kg)
≈ 10.23 m/s
The final velocity (vf) at point B is approximately 10.23 m/s.
Now, we can calculate the final kinetic energy (KEf):
KEf = (1/2) * m * vf^2
= (1/2) * 491 kg * (10.23 m/s)^2
≈ 2.53 x 10^4 J
Therefore, the final kinetic energy (KEf) is approximately 2.53 x 10^4 Joules.
Finally, we can find the change in kinetic energy:
Change in kinetic energy (KEf - KEo) = 2.53 x 10^4 J - 0 J
= 2.53 x 10^4 J
Therefore, the change in the car's kinetic energy from point A to point B is approximately 2.53 x 10^4 Joules.