Why is potential energy vs height a linear relationship when the “end” of the fall happens faster and has less time under gravity?

Question

(Answered, thanks yall) Basically I have three competing understandings: potential energy with respect to height is linear AND gravity is constant in force applied per time (right?) AND at the end of falls you are losing height faster because greater speed.

So with these three things being my understanding I don’t understand how at the end of a fall (some arbitrary speed) you can lose more height and thus PE per second but be accelerated at the same force. I don’t see how you could expend more PE but not be putting in more energy to acceleration… Where does that extra PE lost by higher speed go? Does it take more energy to accelerate when moving faster? It shouldn’t I think ignoring fancy energy momentum stuff that doesn’t apply at 10 mph lol.

So yeah, I don’t get it. I’d be very grateful is someone could solve this for me. I know I must be missing something but don’t know what. This is a question i’ve argued with my brother about a little and tried to look up a few times but the forum posts I’ve found aren’t exactly my issue I think. I also tried asking some ai and it didn’t see my problem I think. For the record I’m in school for chemistry so not a lay person per se but not well read at all either.

In: Physics

Answer 1

There’s a few ways to look at this. The simplest is that energy comes from force over distance. At the end of the fall, it’s moving faster (so covering more distance) and the force is the same. This means that, while the second half of the fall happens faster, gravity is delivering more power during that time and it balances out.

For any situation where force is constant, moving at higher speeds results in more power. This is why raising engine RPM can often give you more power.

Now, your line of reasoning does apply to *momentum*. The first half of the fall gives the falling object more momentum, because it has more time to push, but because energy increases with the square of momentum the diminishing returns on momentum with drop height equate to a constant and steady supply of energy compared to drop height.