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as I'm beginning to learn the basics of physics, there are some concepts that confuse me.

If I'm finding the minimum speed needed for a ball to roll up a hill, the equation is: (1/2)mv^2=mgh solving for v (there is kinetic energy first, and by the time it gets up the hill there should only be potential energy as it rolls at the minimum speed, just enough to get on top of the hill) As I solve for v, the equation is rearranged to v=√2gh

What is confusing to me is that only height and the g are important for solving the question of minimum speed. (based on equation) Why doesn't distance away from the hill matter? For example if a ball was 200m away from a 4 m hill vs a ball 1 m away from a 4 m hill, the equation to find minimum speed needed to get to the top of the hill is still v= √2gh Isn't there the possibility of it slowing down before it can even get to the hill in the first scenario? Why is distance seemingly negligible?

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We are not considering any resistive forces here, like friction or air resistance so there will be no force that can change the total energy of the ball before it starts ascending the hill. Therefore, all of the Kinetic energy will be converted into potential energy.

Without any resistance or external force, the ball could move forever (in accordance with Netwon's First law of motion).

The case would be different if the question involved friction, then the maximum height would be different. The kinetic energy would be equal to the energy dissipated due to friction and the potential energy of the ball at max. height.

Take note : Firction is not a conservative force.

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  • $\begingroup$ An object in motion will stay in motion unless acted on by another force. The reason Earth has not collided with the Sun is because there is no friction in space. Applying the same principle, there does not exist another force that would act on the rolling ball over that distance from the hill (ignoring energy conversion over height) in your example. $\endgroup$
    – user140374
    Apr 30 '17 at 11:35

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