A 81.8-kg skier rides a 3300-m-long lift to the top of a mountain. The lift makes an angle of 12.9 ° with the horizontal. What is the change in the skier's gravitational potential energy?
To determine the change in the skier's gravitational potential energy, you need to know the initial and final heights of the skier. The potential energy of an object in a gravitational field is given by the equation PE = mgh, where PE is the potential energy, m is the mass of the skier, g is the acceleration due to gravity, and h is the height.
In this case, the height can be determined using the length of the lift and the angle it makes with the horizontal. Since the lift goes straight up the mountain, the height can be calculated using trigonometry.
The height (h) is equal to the length of the lift (L) multiplied by the sine of the angle (θ). So, h = L * sin(θ).
Given that the mass of the skier (m) is 81.8 kg, the length of the lift (L) is 3300 m, and the angle (θ) is 12.9°, you can now calculate the height.
h = 3300 m * sin(12.9°) = 716.59 m
Now that you have the height, you can calculate the initial and final potential energies. The initial potential energy (PEi) is when the skier is at the bottom of the lift, and the final potential energy (PEf) is when the skier is at the top of the mountain.
PEi = m * g * 0 (since the skier is at ground level, the initial potential energy is zero)
PEf = m * g * h (using the calculated height)
In both equations, g is the acceleration due to gravity, which is approximately 9.8 m/s².
Substituting the known values:
PEf = (81.8 kg) * (9.8 m/s²) * (716.59 m) = 567,536.82 Joules
Therefore, the change in the skier's gravitational potential energy is 567,536.82 Joules.