# A gravity calculusy kind of quesiton

How would i go about writing an equation for the velocity of an object released at a high altitude above the earth which takes into account the sqaure increase in acceleration with displacement (s^2).

I posted it here because i'm aware it's more of a calculus question than physics.

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arildno
Homework Helper
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Hmm, if s is your displacement from the SURFACE, then R+s will be your distance from the centre of Earth, agreed?
(R being the radius of the Earth)
The adjective is calculous by the way..

appreciated.....

p.s. "The adjective is calculous by the way.." - :tongue: i know

HallsofIvy
Homework Helper
Gravitational force is
$$\frac{-GMm}{r^2}$$
where r is the distance from the center of the earth. If your s is height above the surface of the earth, M is the mass of the earth and R is the radius of the earth, then
$$\frac{d^2s}{dt^2}= \frac{-GM}{(R+s)^2}$$

That's for something falling straight down, not an orbit, of course.
The best way to solve that differential equation is to let v= ds/dt, then not that d2s/dt2= dv/dt= (ds/dt)(dv/ds) (chain rule) = vdv/ds
$$v\frac{dv}{ds}= -\frac{GM}{(R+s)^2}$$
so
$$vdv= -\frac{GMds}{(R+s)^2}$$
That should be easy to integrate.

damn it, i had a feeling it was going to be something that simple. Thanks a lot, you've put my mind at rest.

arildno