A roller-coaster car of mass 1.80 103 kg is initially at the top of a rise at point circle a. It then moves 40.8 m at an angle of 50.0° below the horizontal to a lower point circle b.
(a) Find both the potential energy of the system when the car is at points circle a and circle b and the change in potential energy as the car moves from point circle a to point circle b, assuming y = 0 at point circle b.
To solve this problem, we need to calculate the potential energy at points circle a and circle b, and then find the difference between them to determine the change in potential energy.
The potential energy of an object at a certain height is given by the formula:
PE = m * g * h
Where:
- PE is the potential energy
- m is the mass of the object
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
- h is the height of the object above the reference point (in this case, y = 0 at point circle b)
Now let's calculate the potential energy at point circle a:
PE_a = m * g * h_a
Where h_a is the height of the roller-coaster car at point circle a. Since we are not provided with the exact height, we'll need to calculate it.
To do that, we use the distance and the angle below the horizontal. The vertical component of the distance can be calculated using the sine function:
vertical_distance = distance * sin(angle)
Now, the height at point circle a can be found by subtracting the vertical distance from the total height:
h_a = total_height - vertical_distance
Next, we calculate the potential energy at point circle b. Since the reference point is y = 0 at point circle b, the height at point circle b is simply zero. Therefore, the potential energy at point circle b is:
PE_b = m * g * 0 = 0
Finally, we can find the change in potential energy as the car moves from point circle a to point circle b:
delta_PE = PE_b - PE_a
Substitute the values into the equations to find the answers.