Question

A 900 kg roller coaster car starts at point A, then travels 155 ft at 40.0° below the horizontal to point B

(a) Taking point B to be the level where the gravitational potential energy of the car-Earth system is zero, what is the potential energy (in 3) of the system when the car is at points A and B, and the change in potential energy (in 3) as the coaster moves between these points?
at point A ______
at point B ______
change in potential energy ______
(b) Repeat part (a), setting the zero configuration with the car at point A
at point A ______
at point B ______
change in potential energy ______

Answer 1

The change in gravitational potential energy for the roller coaster car is -266,192.1 J when moving from point A to point B in both scenarios; it is positive at point A when B is zero level, and negative at point B when A is zero level.

Step-by-step explanation:

To calculate the change in gravitational potential energy of the roller coaster car, we will use the equation ΔPEg = mgh, where m is mass, g is the acceleration due to gravity, and h is the change in height. Gravity will be taken as 9.8 m/s2. We'll need to convert the 155 ft distance into meters (1 ft = 0.3048 m) and then use the sine function to find the vertical distance (height) traveled since the car is traveling at a 40.0° angle below the horizontal.

Firstly, we convert 155 ft to meters: 155 ft * 0.3048 m/ft = 47.244 meters. Then, we calculate the height change: h = 47.244 m * sin(40°) ≈ 30.245 meters.

(a) Point A is at the top, so setting point B (where the coaster car is level with the ground) as zero gravitational potential energy:

At point A: PEg = mgh = (900 kg)(9.8 m/s2)(30.245 m) ≈ 266,192.1 J

At point B: PEg = 0 J (by definition of the reference level)

Change in potential energy: ΔPEg = 0 J - 266,192.1 J = -266,192.1 J

(b) If we set the zero gravitational potential energy level at point A:

At point A: PEg = 0 J (by this reference level)

At point B: PEg = -mg(-h) = (900 kg)(9.8 m/s2)(-30.245 m) = -266,192.1 J

Change in potential energy: ΔPEg = -266,192.1 J - 0 J = -266,192.1 J

In both scenarios, the absolute change in potential energy is the same, but the signs differ based on the chosen reference level for potential energy.

Answer 2

(a) At point A: 0 J; At point B: 300,221.06 J; Change in potential energy: +300,221.06 J.

(b) At point A: 300,221.06 J; At point B: 0 J; Change in potential energy: -300,221.06 J.

Explanation:

The potential energy at point A is zero in scenario (b) because the zero configuration is set with the car at point A. As the coaster moves from point A to point B, the potential energy at point B is calculated to be 300,221.06 J. This calculation accounts for the change in elevation, considering the gravitational potential energy of the car-Earth system. The change in potential energy, which is -300,221.06 J in scenario (b), is obtained by subtracting the potential energy at point B from the potential energy at point A.

In scenario (a), where point B is considered as the zero level, the potential energy at point B is zero. At point A, the potential energy is again calculated to be zero in this scenario. When the coaster moves from point A to point B, the potential energy increases, resulting in a positive value of 300,221.06 J. This change in potential energy (+300,221.06 J) is derived by subtracting the potential energy at point A from the potential energy at point B, indicating an increase in the system's potential energy.

The calculations illustrate how setting different points as the zero configuration affects the potential energy values at those points and the resulting change in potential energy as the coaster moves between them.

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