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 for this situation, we need to consider the potential energy, kinetic energy, and any other relevant forms of energy.
Potential energy (PE) is given by the equation PE = mgh, where m is the mass of the car, g is the acceleration due to gravity, and h is the height of the hill.
Kinetic energy (KE) is given by the equation KE = (1/2)mv^2, where m is the mass of the car and v is its velocity.
At the top of the hill, the car has no kinetic energy (KE = 0), so all of the initial energy is in the form of potential energy (PE = mgh). At the bottom of the hill, all of the potential energy has been converted to kinetic energy, and there is no more potential energy left (PE = 0).
So, we can set up the conservation of mechanical energy equation as follows:
Initial mechanical energy (Ei) = Final mechanical energy (Ef)
PE + KE = 0 + KE
mgh + 0 = (1/2)mv^2
To solve this equation for the maximum speed of the car (vf) in terms of the height of the hill (yi), we need to eliminate the mass (m) from both sides of the equation. We can do this by dividing both sides by m, which cancels out the mass:
mgh/m + 0 = (1/2)m(v^2)/m
gh = (1/2)v^2
Now, we can rearrange the equation to solve for vf:
2gh = v^2
Taking the square root of both sides:
√(2gh) = √(v^2)
√(2gh) = v
Therefore, the maximum speed of the car (vf) in terms of the height of the hill (yi) is given by:
vf = √(2gh)
This equation shows that the maximum speed of the car at the bottom of the hill depends on the height of the hill, as well as the acceleration due to gravity (g).