(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
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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.
Kinetic energy is (1/2)mv^2, so it makes sense that when you are in free-fall, with velocity increasing linearly (and hence kinetic energy increasing faster than linear) the result is that your gravitational potential energy is decreasing faster than linear with respect to time.
Many people conflate acceleration with velocity.
Some assumptions: near the surface of mass it’s being drawn to(earth in this case), and much less than terminal velocity.
While falling it’s increasing velocity at a constant rate of 9.8m/s. Total velocity at the end is constant(t*9.8). Total energy would equal velocity*mass.
Kinetic energy is based on velocity squared (so doubling your velocity means you quadruple your kinetic energy). So yes, you are losing PE faster as you fall, but the rate at which you’re gaining kinetic energy is also increasing in turn.
Other comments talked about how the formulas are constructed or try to explain it in other terms. But what I want to do is ask a different question – what if indeed the energy depended on time.
So based on our observations, we know how fast an object will hit a ground based on a certain height. But it doesn’t have to be a straight path. If we take a ball and release it on a ramp from the same height, it’ll reach the same speed only horizontally. But in doing that, we can make the path from highest point to lowest point as long as we want. If the energy was dependent on time, then it would keep increasing and the longer the ramp, the faster it’ll exit. The path can be also squiggly or zigzag and because we increased the path, time increases and therefore the overall energy.
But it doesn’t work like that. Apparently the only thing that matters is the starting height and the end height. So time doesn’t factor in it.
We also have a great example of a very very very long path to fall – satellites. Satellites are constantly “falling” to earth, just in the longest path possible. If time was a factor in their energy, then the energy released when they hit the ground would be enormous. But it doesn’t. So our initial observation of only the beginning height and end height matters (barring any external force and friction).
Energy is the potential to do work. Work can be defined as a FORCE over a distance. If you exert a force of 5 pounds (lbf) on an object and move it 5 feet, you have done (5×5=) 25 ft-lbf of work. So the only two factors in how much potential energy is stored in an object at some height are: force and distance. Distance is jsut the height, sot hats easy. But what do we do with the force term? WEll, how much force is the object exerting? Well, luckily we have Newtons Law to help us out, which states the F=ma (Force – mass * acceleration). So we cna figure out the force exerted on the objectg from gravity if we know its mass, and we know the acceleration due to gravity. So now we know that we can figure out the potential energy if we know the mass of the object, its height above the datum, and the acceleration due to gravity! Gool thing is, gravity is a constant (assuming youre staying on/near the surface of teh earth), the “a” doesn;t really change if youre at 1 ft or 1,000 ft of elevation.
POut all that together, and you can see that the only things that determine an object potential energy are its mass, its height above the datum, and the acceleration due to gravity. Nothing to do with its speed. If its at rest at 1,000 ft, or moving 50 ft/s downward at 1,000 ft, it has the same potential energy in that instant. If its at 50 feet, it will have another amount, regardless of its velocity. The only factors that come into play are mass, acceleration, and height. And since mass and acceleration dont change during a fall, the only variable that influences PE is the height. And since height scales linearly with height, that means the PE will scale linearly with height.
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