A skateboarder starts at rest at the top of a ramp. When he reaches the bottom of the ramp, he has a total kinetic energy of 1,646 joules. How much potential energy did he start with at the top of the ramp, if energy was fully conserved?

Arunner is training for an upcoming marathon by running around a 100-m-diameter circular track at constant speed. let a coordinate system have its origin at the center of the circle with the x-axis pointing east and the y-axis north. the runner starts at (x, y) = (50 m, 0 m) and runs 2.5 times around the track in a clockwise direction.

Mechanical energy is when the amount of kinetic and potential energies added together remains the same. conserved created lost

Suppose that f : rn → rm and that a ∈ k, where k is a connected subset of rn . suppose further that for each x ∈ k there exists a δx > 0 such that f(x) = f(y) for all y ∈ bδx (x). prove that f is constant on k; that is, f(x) = f(a) for all x ∈ k