An 800.0 kg roller coaster car is at rest at the top of a 95 m hill. It rolls down the first drop to a height of 31 m. When it travels to the top of the second hill, it is moving at 28 m/s. It then rolls down the second hill until it is at ground level.

What is the kinetic and potential energy at the top and bottom of each hill?

1.

PE1 = mgh1 = 800•9.8•95 =744800 J.
KE1=0.
Total E1 = PE1+ KE1=744800 J.
2.
PE2 = 800•9.8•31=243040
PE1= PE2+KE2
KE2 = PE1- PE2 =
=744800 - 243040=501760 J.
Total E2 = PE2+ KE2=744800 J.
3.
KE3 =mv²/2= 800•(28)²/2 =313600 J.
PE3 =744800-313600 = 431200 J.
Total E3 = PE3+ KE3=744800 J.
4.
PE4=0.
KE4 = 744800 J.
Total E4 = PE4+ KE4=744800 J.

They presumably want to to assume total mechanical energy (kinetic + potential) is constant, even though that is not the case.

At the top of the 95 m hill, all of the energy is potential, and equals
M g H = 744,800 J. Use the height or velocity at the other location, and the total energy (748,800 J) , to determine kinetic and potential energies

To find the kinetic and potential energy at the top and bottom of each hill, we need to use the formulas for kinetic energy (KE) and potential energy (PE). The formulas are:

KE = (1/2) * m * v^2
PE = m * g * h

Where:
KE is the kinetic energy
m is the mass of the roller coaster car (800.0 kg)
v is the velocity of the roller coaster car
PE is the potential energy
g is the acceleration due to gravity (9.8 m/s^2)
h is the height of the roller coaster car

Let's calculate the kinetic and potential energy at each point:

1. Top of the first hill (initial position):
At the top of the hill, the roller coaster car is at rest, so its velocity is 0.

KE = (1/2) * 800.0 kg * 0^2 = 0 J
PE = 800.0 kg * 9.8 m/s^2 * 95 m = 752,800 J

2. Bottom of the first hill:
The roller coaster car reaches a height of 31 m at the bottom of the first hill.

KE = (1/2) * 800.0 kg * v^2
PE = 800.0 kg * 9.8 m/s^2 * 31 m

To find the velocity (v) at the bottom of the hill, we can use the conservation of energy principle, which states that the total mechanical energy (the sum of kinetic and potential energy) remains constant. Therefore, the initial potential energy at the top is equal to the final kinetic energy at the bottom:

KE (bottom) = PE (top)

(1/2) * 800.0 kg * v^2 = 752,800 J

Solving for v, we find:

v = sqrt(2 * (752,800 J) / (800.0 kg)) ≈ 30.05 m/s

So,

KE (bottom) = (1/2) * 800.0 kg * (30.05 m/s)^2 ≈ 360,563 J
PE (bottom) = 800.0 kg * 9.8 m/s^2 * 31 m ≈ 241,360 J

3. Top of the second hill:
The roller coaster car reaches a height of 0 m at the top of the second hill and is moving with a velocity of 28 m/s.

KE = (1/2) * 800.0 kg * (28 m/s)^2 ≈ 313,600 J
PE = 800.0 kg * 9.8 m/s^2 * 0 m = 0 J

4. Bottom of the second hill (ground level):
The roller coaster car reaches a height of 0 m at the bottom of the second hill.

KE = (1/2) * 800.0 kg * v^2
PE = 800.0 kg * 9.8 m/s^2 * 0 m

Using the same conservation of energy principle as before:

KE (bottom) = (1/2) * 800.0 kg * (28 m/s)^2 ≈ 313,600 J
PE (bottom) = 800.0 kg * 9.8 m/s^2 * 0 m = 0 J

So, the kinetic and potential energy at each point are:

First hill:
Top: KE = 0 J, PE ≈ 752,800 J
Bottom: KE ≈ 360,563 J, PE ≈ 241,360 J

Second hill:
Top: KE ≈ 313,600 J, PE = 0 J
Bottom: KE ≈ 313,600 J, PE = 0 J

To determine the kinetic and potential energy at the top and bottom of each hill, we need to use the equations for potential energy (PE) and kinetic energy (KE).

The formula for potential energy is:
PE = mgh

where m represents the mass, g is the acceleration due to gravity (9.8 m/s^2), and h is the height.

The formula for kinetic energy is:
KE = (1/2)mv^2

where m is the mass and v is the velocity.

Let's calculate the energies at each position:

1. Top of the first hill (height = 95 m):
- Potential energy (PE) = 800 kg * 9.8 m/s^2 * 95 m
- As the roller coaster is at rest, the kinetic energy (KE) is zero.

2. Bottom of the first hill (height = 31 m):
- Potential energy (PE) = 800 kg * 9.8 m/s^2 * 31 m
- Kinetic energy (KE) = (1/2) * 800 kg * (velocity)^2
To find the velocity, we can use the equation for conservation of energy:
PE (top) = PE (bottom) + KE (bottom)

3. Top of the second hill (height = 0 m, ground level):
- Potential energy (PE) is zero.
- Kinetic energy (KE) = (1/2) * 800 kg * 28 m/s

4. Bottom of the second hill (height = 0 m, ground level):
- Potential energy (PE) is zero.
- Kinetic energy (KE) = (1/2) * 800 kg * (velocity)^2
To find the velocity, we can use the equation for conservation of energy:
KE (top) = PE (bottom) + KE (bottom)

By calculating the potential energy and kinetic energy using the given formulas and the provided data, you can find the values for each position.