Aball starts at the top of a ten-story building, falls straight down, and hits the ground with a velocity bottom. how does that velocity change if it falls down a ramp at an angle of 45 degrees? how does that velocity change if it falls down a ramp that goes down, around, over, left, right, upside-down, makes a loop, goes up, down, and finally reaches the ground?

velocity at bottom of the trajectory is always the same.

Explanation:

As the gravitational field is a conservative one, the work done by the gravitational force is independent of the trajectory, and only  depends on the height.

As the work is equal to the change in kinetic energy, for a given height, the final speed will always the same, as it must be satisfied the following equation:

ΔK = m . g. h

1/2 m v² = m .g .h⇒ v =√2.g.h

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