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How can I calculate the maximum height of a projectile that is launched from the surface of the earth with a given initial velocity? (ignoring air resistance in the atmosphere)

I understand how to solve this type of question when the acceleration is constant (close to the surface of the earth). However, I don't know how to when $g$ is changing.

Do I integrate $g = Gm/r^2$?

I want to solve the question without using conservation of energy.

Qmechanic
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Blugh Bla
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1 Answers1

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Solve the differential equation

$$\frac{d^2r}{dt^2}=-\frac{GM}{r^2}$$

for the function $r(t)$ with the initial conditions

$$r|_{t=0}=R$$

and

$$\frac{dr}{dt}\bigg|_{t=0}=V.$$

Then find the maximum of $r(t)$. The maximum height is $r_\text{max}-R$.

G. Smith
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