Maximum height of a projectile when $g$ is not constant

Question

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.

What would you be integrating when you do this calculation? Are you simply looking for the average value of $g$? — Quantum Mechanic, Jun 01 '21 at 18:42
is it thrown radially upward? or at an angle from earth's surface? — Rishab Navaneet, Jun 01 '21 at 18:46
You could use Lagrangian mechanics, which does not depend on conservation of energy. Instead, conservation of energy is a consequence of a Lagrangian that does not explicitly depend on time. — gandalf61, Jun 01 '21 at 19:40

score 1 · Accepted Answer · answered Jun 01 '21 at 18:55

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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 $r_\text{max}-R$.

answered Jun 01 '21 at 18:55

G. Smith

51,534

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Note that the analytic solution to this ODE amounts to applying conservation of energy anyhow. – Michael Seifert Jun 01 '21 at 19:04
@G.Smith please show how to solve $F=m a$ analytically without using conservation of energy first (at least the differential form). You are faced with problem #2 in why we need calculus to solve varying acceleration problems. – JAlex Jun 01 '21 at 19:10
@G. Smith you could add that analytical solutions can be found by multiplying both sides with $\frac{dr}{dt}$. Even though it ultimately is the energy conservation equation, it wasn't specifically used – Rishab Navaneet Jun 01 '21 at 19:13
@RishabNavaneet That would be getting too close to providing a complete solution to a homework problem. – G. Smith Jun 01 '21 at 19:15
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I would quibble with the idea that you can solve this problem while remaining ignorant of the existence of conserved quantities. I'm unaware of any technique for analytically solving this equation that doesn't involve noticing that $\frac{1}{2} \dot{r}^2 - GM/r$ is a first integral of the motion. – Michael Seifert Jun 01 '21 at 19:16
Someone who doesn’t even know what “a conserved quantity” means can find the trajectory and maximize it. – G. Smith Jun 01 '21 at 19:31

Maximum height of a projectile when $g$ is not constant

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