A rollercoaster car of mass 600.0 kg is moving 2.4m/s at the top of a peak that is 25m off the ground. The car moves down slope and up the next hill. Assume that it loses 20,000J of energy by the next time it gets to the top of the next peak due to friction forces. If the speed of the car must be at a minimum of 2.0m/s to keep it moving on to the next slope, what is the maximum height allowed for the 2nd peak.
The initial total(kinetic + potential) energy, with respect to ground level zero P.E., is
(1/2) M V^2 + M g H = 1728 J + 147,150 J = 148,778 J. If it loses 20,000 J before reaching the next peak, and has 2.0 m/s velocity there, then
128,778 = (1/2) M V^2 + M g H'
= 1200 + M g H'
Solve for the new maximum height, H'. I get H' = 21.7 m.
I used 9.81 m/s^2 for g. No guarantees here.. check my assumptions and calculations.
To solve this problem, we'll use the conservation of energy principle. The initial total mechanical energy of the rollercoaster at the top of the first peak is equal to the sum of its potential and kinetic energy:
Initial Energy (Ei) = Potential Energy (PE) + Kinetic Energy (KE)
The potential energy is given by the equation:
Potential Energy (PE) = mass * gravity * height
where
mass = 600.0 kg (mass of the rollercoaster)
gravity = 9.8 m/s^2 (acceleration due to gravity)
height = 25 m (height of the first peak)
The kinetic energy is given by the equation:
Kinetic Energy (KE) = 0.5 * mass * velocity^2
where
mass = 600.0 kg (mass of the rollercoaster)
velocity = 2.4 m/s (velocity of the rollercoaster at the top of the peak)
Therefore, the initial energy can be calculated as:
Ei = PE + KE
= (mass * gravity * height) + (0.5 * mass * velocity^2)
Next, we'll calculate the final energy (Ef) of the rollercoaster at the top of the next peak. We know that 20,000 J of energy is lost due to friction forces, so we subtract this loss from the initial energy:
Ef = Ei - Energy Lost
= Ei - 20,000 J
Now, to find the maximum height allowed for the second peak, we'll calculate the potential energy at that height and set it equal to the final energy. This is because the rollercoaster must have enough energy to move up the next slope with a minimum speed of 2.0 m/s.
Potential Energy (PE) = mass * gravity * height
Therefore, we have the equation:
Ef = PE + KE
= (mass * gravity * height) + (0.5 * mass * velocity^2)
Solving for height:
(mass * gravity * height) + (0.5 * mass * velocity^2) = Ef
Substituting the values we know:
(600.0 kg * 9.8 m/s^2 * height) + (0.5 * 600.0 kg * 2.0 m/s)^2 = Ef
Solving this equation will give us the maximum height allowed for the second peak.