# Maximum Height Calculator – Projectile Motion

Created by Hanna Pamuła, PhD
Reviewed by Bogna Szyk and Steven Wooding
Last updated: Dec 21, 2022

The maximum height calculator is a tool for finding the maximum vertical position of a launched object in projectile motion. Whether you need the max height formula for an object starting directly off the ground or from some initial elevation – we've got you covered. If you're still wondering how to find the maximum height of a projectile, read the two short paragraphs below, and everything should become clear.

## How to find the maximum height of a projectile?

The maximum height of the object is the highest vertical position along its trajectory. The object is flying upwards before reaching the highest point – and it's falling after that point. It means that at the highest point of projectile motion, the vertical velocity is equal to $0$ ($v_y = 0$).

$\small 0 = v_y\! -\! g \cdot t = v_0 \cdot \sin(\alpha) - g \cdot t_{\mathrm{h}}$

From that equation we can find the time $t_{\mathrm{h}}$ needed to reach the maximum height $h_{\mathrm{max}}$:

$\small t_{\mathrm{h}} = v_0\cdot\frac{\sin(\alpha)}{g}$

The formula describing vertical distance is:

$\small y = v_y\cdot t - g\cdot \frac{t^2}{2}$

So, given $y = h_{\mathrm{max}}$ and $t = t_{\mathrm{h}}$, we can join those two equations together:

$\small \begin{split} &h_\mathrm{max} = v_0\cdot t_\mathrm{h} - g\cdot\frac{t_\mathrm{h}^2}{2}\\[1em] &=v_0^2\cdot \frac{\sin^2(\alpha)}{g} - g\cdot\frac{\left(v_0\cdot\frac{\sin(\alpha)}{g}\right)^2}{2}\\[1em] &=v_0^2\cdot \frac{\sin^2(\alpha)}{2\cdot g} \end{split}$

And what if we launch a projectile from some initial height $h$? No worries! Apparently, the calculations are a piece of cake – all you need to do is add this initial elevation!

$\small h_\mathrm{max}= h+\frac{v_0^2\cdot \sin(\alpha)}{2\cdot g}$

Let's discuss some special cases with changing angle of launch:

• If $\alpha = 90\degree$, then the formula simplifies to:
$\small\qquad h_{\mathrm{max}} = h+\frac{v_0^2}{2\cdot g}$

And the time of flight is the longest.

If, additionally, $v_y = 0$, then it's the case of free fall, which we detailed at the free fall calculator. Also, you may want to have a look at our even more accurate equivalent – the free fall with air resistance calculator.

• If $\alpha = 45\degree$, then the equation may be written as:
$\small\qquad h_{\mathrm{max}} = h+\frac{v_0^2}{4\cdot g}$

And in that case, the range is maximal if launching from the ground ($h = 0$).

• If $\alpha = 0\degree$, then vertical velocity is equal to $0$ ($v_y = 0$). In this case, we can calculate the horizontal projectile motion. As the sine of $0\degree$ is $0$, then the second part of the equation disappears, and we obtain:
$\small\qquad h_\mathrm{max} = h$

The initial height from which we're launching the object is the maximum height in projectile motion.

The motion of a projectile is a classic problem in physics, and it has been analyzed in every possible aspect. The fact that we can easily reproduce it and observe it was a contributing factor. We decided to create a suit of tools related to the motion of a projectile:

## Maximum height calculator helps you find the answer

Just relax and look how easy-to-use this maximum height calculator is:

1. Choose the velocity of the projectile. Let's type $30\ \mathrm{ft/s}$.

2. Enter the angle. Assume we're kicking a ball ⚽ at an angle of $70\degree$.

3. Optionally, type the initial height. In our case, our starting position is the ground, so type in $0$. Can the ball fly over a $13\ \mathrm{ft}$ fence?

4. Here it is – maximum height calculator displayed the answer! It's $12.35\ \mathrm{ft}$. So it will not fly over the mentioned barrier – throw it harder or increase the angle to reach your goal.

Just remember that we don't take air resistance into account!

Hanna Pamuła, PhD
Velocity
ft/s
Angle of launch
deg
Initial height
ft
Maximum height
Maximum height
ft
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