At the beginning of a roller coaster ride, the car is lifted to the top of a large hill and
released. The speed of the car at the top of the hill is small, so we will assume it to be
zero. The car rolls freely down this hill and reaches its maximum speed at the bottom.
If the roller coaster were frictionless, mechanical energy would be conserved… Ei = Ef.
Showing all terms for potential and kinetic energy, set up the conservation of mechanical
energy for this situation…
yi
vf
Solve this relationship for the maximum speed of the car, vf, in terms of height, yi .
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See previous post: Tue,12-3-13,10:36 PM.
To set up the conservation of mechanical energy equation, we need to consider the initial and final states of the roller coaster.
The initial state is when the car is lifted to the top of the hill. At this point, the car is not moving, so its kinetic energy (KEi) is zero. The potential energy (PEi) is given by the equation PEi = m * g * yi, where m is the mass of the car, g is the acceleration due to gravity, and yi is the height of the car above a reference point.
The final state is when the car reaches the bottom of the hill, where it has its maximum speed. At this point, the car has kinetic energy (KEf) and potential energy (PEf). Since the roller coaster is frictionless, no energy is lost due to friction, so the total mechanical energy is conserved.
So, the conservation of mechanical energy equation, Ei = Ef, can be written as:
KEi + PEi = KEf + PEf
Substituting the values for initial and final states, we get:
0 + m * g * yi = 0.5 * m * vf^2 + 0
Since the initial speed is assumed to be zero, the initial kinetic energy is zero. The final potential energy is zero at the bottom of the hill, as the car is at the reference point. Simplifying the equation, we get:
m * g * yi = 0.5 * m * vf^2
Now, we can solve this equation for the maximum speed of the car, vf, in terms of the height, yi:
vf^2 = (2 * g * yi)
Taking the square root of both sides, we get:
vf = √(2 * g * yi)
So, the maximum speed of the car, vf, in terms of the height, yi, is given by the equation:
vf = √(2 * g * yi)