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Níckolas Alves
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Falling to Earth from Orbit

I apologize if this question seems incredibly remedial. Most likely, I am just missing something obvious. All of that said, I am an honors math major at a Top-20 university and I got a 5 on AP Physics, and yet I cannot crack this.

My question is this: suppose we ascend to a height $$h$$ away from the surface of the earth and drop an object. High school physics tells us that it will accelerate at a speed of $$9.8 \frac ms$$ and will hit the ground in $$\sqrt{\frac h{4.9}}$$ seconds. However, if we ascend high enough, the gravitational force from the earth is less, and so the acceleration is less, and it will take longer to hit the ground. What is the true equation for how far the object has fallen and how long it will take to reach the ground?

Here's what I have so far: for generality, let the earth or other planet have a radius of $$r$$ and surface acceleration due to gravity of $$G$$.

We can derive from the variables above that the force acting on the object at height $$h$$ is $$mG\left(\frac r h\right)^2$$. Thus it will accelerate at $$G\left(\frac r h\right)^2 = Gr^2 \frac 1 {h^2}\frac m{s^2}$$. I want to integrate this with respect to time, and yet my variable is in meters. I need some related rate between the two, but that related rate would be how many seconds it takes to fall some number of meters, which is exactly what I am looking for to begin with. Any advice/the answer? If there's a cool equation using relativity instead of Newtonian mechanics, I would appreciate that as well.