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 00 (vy=0v_y = 0).

0=vy ⁣ ⁣gt=v0sin(α)gth\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 tht_{\mathrm{h}} needed to reach the maximum height hmaxh_{\mathrm{max}}:

th=v0sin(α)g\small t_{\mathrm{h}} = v_0\cdot\frac{\sin(\alpha)}{g}

The formula describing vertical distance is:

y=vytgt22\small y = v_y\cdot t - g\cdot \frac{t^2}{2}

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

hmax=v0thgth22=v02sin2(α)gg(v0sin(α)g)22=v02sin2(α)2g\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 hh? No worries! Apparently, the calculations are a piece of cake – all you need to do is add this initial elevation!

hmax=h+v02sin(α)2g\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 α=90°\alpha = 90\degree, then the formula simplifies to:
hmax=h+v022g\small\qquad h_{\mathrm{max}} = h+\frac{v_0^2}{2\cdot g}

And the time of flight is the longest.

If, additionally, vy=0v_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 α=45°\alpha = 45\degree, then the equation may be written as:
hmax=h+v024g\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=0h = 0).

  • If α=0°\alpha = 0\degree, then vertical velocity is equal to 00 (vy=0v_y = 0). In this case, we can calculate the horizontal projectile motion. As the sine of 0°0\degree is 00, then the second part of the equation disappears, and we obtain:
hmax=h\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 ft/s30\ \mathrm{ft/s}.

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

  3. Optionally, type the initial height. In our case, our starting position is the ground, so type in 00. Can the ball fly over a 13 ft13\ \mathrm{ft} fence?

  4. Here it is – maximum height calculator displayed the answer! It's 12.35 ft12.35\ \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
Projectile motion: maximum height
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