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 .
At top of hill:
KE + PE = mg*h
0 + PE = mg*h
PE = mg*h
At bottom of hill:
KE + PE = mg*h
KE + 0 = mg*h
KE = mg*h = 0.5m*V^2
0.5m*V^2 = mg*h
V^2 = 2g*h
V = Sqrt(2g*h).
V = Sqrt(19.6*h).
Sqrt means Square root.
To set up the conservation of mechanical energy, we need to consider the potential energy and kinetic energy of the system.
At the top of the hill, the car has potential energy (PE) due to its height. Let's denote this initial potential energy as PEi.
At the bottom of the hill, when the car reaches its maximum speed, it has converted all its potential energy to kinetic energy (KE). Let's denote this final kinetic energy as KEf.
Since the roller coaster is assumed to be frictionless, mechanical energy is conserved. This means that the total initial mechanical energy (Ei) is equal to the total final mechanical energy (Ef).
For this situation, the only forms of energy involved are potential energy and kinetic energy. Therefore, the conservation of mechanical energy equation can be expressed as:
PEi = KEf
Now, let's express the potential energy and kinetic energy in terms of the given variables.
Potential energy (PE) is given by the equation:
PE = m * g * h
Where:
m is the mass of the car (which cancels out in the conservation of energy equation since it appears in both sides)
g is the acceleration due to gravity (which is approximately 9.8 m/s^2)
h is the height of the hill (denoted as yi in the question)
So, the potential energy at the top of the hill (PEi) is:
PEi = m * g * yi
The kinetic energy (KE) is given by the equation:
KE = 0.5 * m * v^2
Where:
v is the velocity/speed of the car at the bottom of the hill (denoted as vf in the question)
For the final kinetic energy (KEf), this equation becomes:
KEf = 0.5 * m * vf^2
Plugging these expressions for potential energy and kinetic energy into the conservation of mechanical energy equation:
m * g * yi = 0.5 * m * vf^2
Simplifying and canceling out the mass (m) on both sides:
g * yi = 0.5 * vf^2
Now, solving this equation for the maximum speed of the car (vf):
vf^2 = 2 * g * yi
Taking the square root of both sides gives:
vf = √(2 * g * yi)
Therefore, the maximum speed of the car (vf) in terms of the height (yi) is given by:
vf = √(2 * g * yi)