# Maximum Height Calculator – Projectile Motion

Created by Hanna Pamuła, PhD
Reviewed by Bogna Szyk and Steven Wooding
Last updated: Jun 05, 2023

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!

## FAQ

### How do I find the maximum height of a ball thrown up?

To find the maximum height of a ball thrown up, follow these steps:

1. Write down the initial velocity of the ball, v₀.
2. Write down the initial height, h₀.
3. Replace both in the following formula: h_max = h₀ +(v₀)²/ 2g where g is the acceleration due to gravity, g ~ 9.8 m/s².

### What angle has the longest flight time?

90°. You can determine the flight time (t) with the formula t = 2 × V × sin(α) / g. For a given velocity V, this function reaches its maximum value when sin (α) = 1, which occurs at α = 90°.

### What factors affect projectile motion?

The main factor that could affect the motion of a projectile is friction. When air resistance is introduced, the friction opposes the direction of movement, decreasing the velocity components of the projectile.

### Does the mass of the projectile affect maximum height?

No. The maximum height a projectile can reach after being thrown is only affected by the initial velocity and the launch angle. The mass of the projectile does not affect the maximum height.

Hanna Pamuła, PhD
Velocity
ft/s
Angle of launch
deg
Initial height
ft
Maximum height
Maximum height
ft People also viewed…

### Black hole collision

The Black Hole Collision Calculator lets you see the effects of a black hole collision, as well as revealing some of the mysteries of black holes, come on in and enjoy!

### Flat vs. round Earth

Omni's not-flat Earth calculator helps you perform three experiments that prove the world is round.

### Length contraction

Use the length contraction calculator to discover that length is relative to the point of view.

### Snell's law

Use the Snell's law calculator to analyze the refraction of a ray.