A roller coaster with a mass of 500.0 kg starts from a height of 27.0 meters with a speed of 5.2 m/s. It descends to a height of 2.00 meters before it starts back up. Some energy is loss to the work due to friction. The speed at the bottom of the ramp is 22.0 m/s. How much energy was lost to friction?
1/2 m v^2 + m g h - 1/2 m V^2
500 [(1/2 * 5.2^2) + (9.8 * 25) - (1/2 * 22)] ... Joules
To find the amount of energy lost to friction, we first need to calculate the initial potential energy, final potential energy, and final kinetic energy of the roller coaster.
The initial potential energy (PEi) is given by the formula:
PEi = mass * g * height
where mass is the mass of the roller coaster (500.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and height is the initial height (27.0 m). Plugging in these values, we get:
PEi = 500.0 kg * 9.8 m/s^2 * 27.0 m = 132,300 J
The final potential energy (PEf) is given by the formula:
PEf = mass * g * height
where height is the final height (2.00 m). Plugging in the values, we get:
PEf = 500.0 kg * 9.8 m/s^2 * 2.00 m = 9,800 J
The final kinetic energy (KEf) is given by the formula:
KEf = 0.5 * mass * velocity^2
where mass is the mass of the roller coaster (500.0 kg) and velocity is the final velocity (22.0 m/s). Plugging in the values, we get:
KEf = 0.5 * 500.0 kg * (22.0 m/s)^2 = 110,000 J
According to the law of conservation of energy, the initial potential energy should be equal to the sum of the final potential energy and final kinetic energy:
PEi = PEf + KEf
Therefore, the energy lost to friction can be calculated as:
Energy lost = PEi - (PEf + KEf)
Plugging in the values, we get:
Energy lost = 132,300 J - (9,800 J + 110,000 J) = 12,500 J
Therefore, the energy lost to friction is 12,500 J.