a skaterboarder is moving 5.25 m/s when he starts to roll up a frictionless hill. how much higher is he when he comes to a stop? (energy problem)
To determine the height gained by the skateboarder, we can use the conservation of mechanical energy. According to the law of conservation of energy, the total mechanical energy of a system remains constant if no external forces (such as friction) are acting on it.
In this case, we can assume that only gravitational potential energy and kinetic energy are involved.
The initial mechanical energy of the skateboarder is the sum of his kinetic energy and potential energy. At the top of the hill, when he comes to a stop, all of his initial kinetic energy will be converted into potential energy. Therefore, the change in potential energy will equal his initial kinetic energy.
The formula for kinetic energy (KE) is KE = 1/2 * mass * velocity^2, where the mass is not given. However, since we only need to find the height difference, we can calculate the change in potential energy (ΔPE) without knowing the mass.
The formula for potential energy (PE) is PE = mass * gravitational acceleration * height.
Since the skateboarder's final velocity is zero when he comes to a stop, the change in potential energy (∆PE) can be calculated as follows:
∆PE = KE (initial) = 1/2 * mass * velocity^2
Substituting the given values:
∆PE = 1/2 * mass * (5.25 m/s)^2
To determine the height difference (∆h), we rearrange the potential energy equation as follows:
∆h = ∆PE / (mass * gravitational acceleration)
Since the problem states that there is no friction, the gravitational acceleration (g) can be taken as 9.8 m/s^2.
∆h = (∆PE) / (mass * 9.8 m/s^2)
Since we don't have the mass of the skateboarder, we can cancel it out while solving the equation. Therefore, the height difference (∆h) is independent of mass.
∆h = (∆PE) / (9.8 m/s^2)
Calculating ∆PE:
∆PE = 1/2 * mass * (5.25 m/s)^2 = 1/2 * mass * 27.5625 m^2/s^2
Substituting ∆PE:
∆h = (1/2 * mass * 27.5625 m^2/s^2) / (9.8 m/s^2)
Now we can calculate the height difference (∆h) using the given formula and solve for ∆h.