Answer :
The potential energy of the roller coaster car-Earth system at point A is 0 J, and at point B it is -1.47 × 10⁵ J. The change in potential energy as the coaster moves between these points is -1.47 × 10⁵ J.
The potential energy of an object in a gravitational field is given by the equation PE = mgh, where m is the mass of the object, g is the acceleration due to gravity, and h is the height or vertical position of the object.
At point A, the potential energy of the roller coaster car-Earth system is zero because we are taking point B as the reference level where the potential energy is defined to be zero. Therefore, the potential energy at point A is 0 J.
To calculate the potential energy at point B, we need to determine the height of the roller coaster car above the reference level. We can use trigonometry to find the vertical component of the displacement from point A to point B. The vertical displacement is given by Δy = 125 ft × sin(40°) = 80.3 ft.
Next, we convert the vertical displacement to meters (1 ft = 0.3048 m) and multiply it by the mass of the roller coaster car and the acceleration due to gravity (9.8 m/s²) to calculate the potential energy at point B:
PE_B = mgh = 1100 kg × 9.8 m/s² × 24.4 m = -1.47 × 10⁵ J
The negative sign indicates that the potential energy at point B is below the reference level.
Finally, the change in potential energy as the coaster moves between points A and B is given by the difference in potential energy:
ΔPE = PE_B - PE_A = (-1.47 × 10⁵ J) - (0 J) = -1.47 × 10⁵ J
Therefore, the change in potential energy as the coaster moves between points A and B is -1.47 × 10⁵ J.
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Question : A 1100kg roller coaster car starts at point A, then travels 125ft at 40.0 degrees 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 J) of the system when the car is at points A and B, and the change in potential energy (in J) as the coaster moves between these points?
The roller coaster car starts at point A and travels 165 feet to point B, at a 40.0° angle below the horizontal. Let's break down the problem step-by-step.
1. First, we need to determine the horizontal and vertical components of the displacement. Since the roller coaster car is traveling at an angle below the horizontal, we need to find the horizontal and vertical distances separately.
2. The horizontal displacement can be found using the cosine function. The formula for horizontal displacement is given by:
horizontal displacement = distance * cos(angle)
Plugging in the values, we get:
horizontal displacement = 165 ft * cos(40.0°)
3. The vertical displacement can be found using the sine function. The formula for vertical displacement is given by:
vertical displacement = distance * sin(angle)
Plugging in the values, we get:
vertical displacement = 165 ft * sin(40.0°)
4. Now, we need to determine the final position of the roller coaster car at point B. To do this, we need to add the horizontal and vertical displacements to the initial position at point A.
Let's assume the initial position at point A is (0, 0). Adding the horizontal displacement to the x-coordinate and the vertical displacement to the y-coordinate, we get the final position at point B.
So, the coordinates of point B are:
x-coordinate = 0 + horizontal displacement
y-coordinate = 0 + vertical displacement
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