# Horizontal Projectile Motion Calculator

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

This horizontal projectile motion calculator is a tool to solve a particular case of projectile motion, where an object is launched horizontally from an elevated plane. Type in any two values, and the rest will be calculated in a blink of an eye! Furthermore, the trajectory will be displayed below the results.

Keep scrolling to find out the horizontal projectile motion equations and a simple example of calculations – you shouldn't have problems with your exercises after reading our explanation!

## Horizontal projectile motion equations

As we said, horizontal projectile motion equations are a particular case of general formulas. We don't need to specify the launch angle, as it's parallel to the ground (so the angle is equal to 0°). As a result, we have only one component of initial velocity – $V_x = V$, whereas $V_y = 0$.

We will take the starting point to be at the origin. Then, the equations of motion can be written as:

Distance

• Horizontal distance can be expressed as $x = V t$.
• Vertical distance from the ground is described by the formula $y = – \frac{1}{2}g t^2$, where $g$ is the gravity acceleration, and $h$ is an elevation.

Velocity

• Horizontal velocity is equal to $V$.
• Vertical velocity can be expressed as $–g t$.

Acceleration

• Horizontal acceleration is equal to 0.
• Vertical acceleration is equal to $-g$ (because only gravity acts on the projectile).

The horizontal projectile motion equations look as follows:

Equation of a trajectory

We can combine the equations $x = V t$ and $y = – \frac{1}{2}g t^2$ to get rid of $t$. The trajectory is then equal to:

$y = – \frac{1}{2}g t^2 = \frac{-gx^2}{2V^2}$

If you compare it with the equation in the trajectory calculator, you can appreciate how much simplification there is for a horizontal motion!

Time of flight

To find the time of flight of the projectile, we need to calculate when the projectile hits the ground. In our coordinate system, it happens when the $y$ coordinate is equal to $h$: $\frac{1}{2}g t^2 = h$. From that equation, we can find that the time of flight equals:

$t = \sqrt{\frac{2h}{g}}$

Range of the projectile

The range of the projectile is the total horizontal distance traveled in the flight time. Then, we can write down the equation as:

$r = Vt = v\sqrt{\frac{2h}{g}}$

Again, this formula would be more complicated if the angle weren't set to 0°. If you're curious to see it, check the projectile range calculator.

We won't calculate the maximum height here (see maximum height calculator instead), as we don't have an initial vertical velocity component – and that means that the maximal height is the one from which we're starting.

In all calculations, we neglected the air resistance acting on the projectile. Thus, the sum of kinetic and potential energies is conserved. You can read more about the latter in our potential energy calculator.

## Example of horizontal projectile motion calculations

Let's assume we want to calculate the time of flight and distance traveled by a ball ⚽ thrown from the Eiffel tower with a horizontal speed only, e.g., 7 m/s.

1. Enter the velocity. In our case, it's 7 m/s. Change the units if needed.

2. Type in the initial height from which the motion starts. The Eiffel tower is 324 meters (1,063 ft) tall, but the upper platform is 276 m (906 ft) above the ground. So, let's type 276 m into the proper box.

3. Our horizontal projectile motion calculator shows the time of flight, distance, and trajectory! We found out that it takes 7.5 seconds for the ball to reach the ground and that the horizontal displacement is ~52.52 m. Awesome!

Remember that our tools are really flexible: just type any two values, and the horizontal projectile motion calculator will do its job. You can, for example, check out what velocity is needed to throw the ball to reach a distance of 100 m from the base of a tower.

## FAQ

### How do I calculate the horizontal distance in projectile motion?

To calculate the horizontal distance in projectile motion, follow the given steps:

1. Multiply the vertical height h by 2 and divide by acceleration due to gravity g.

2. Take the square root of the result from step 1 and multiply it with the initial velocity of projection u to get the horizontal distance.

3. You can also multiply the initial velocity u with the time taken by the projectile to reach the ground t to get the horizontal distance.

### How do I calculate the time of flight in horizontal projectile motion?

To calculate the time of flight in horizontal projectile motion, proceed as follows:

1. Find out the vertical height h from where the projectile is thrown.

2. Multiply h by 2 and divide the result by g, the acceleration due to gravity.

3. Take the square root of the result from step 2, and you will get the time of flight in horizontal projectile motion.

### Is there horizontal acceleration in projectile motion?

No, there is no horizontal acceleration in projectile motion. The velocity of a projectile is constant in the horizontal direction. Hence, the acceleration is also zero along the horizontal direction.

### What is the vertical acceleration of a projectile when it is projected horizontally?

9.8 m/s². A projectile thrown horizontally moves under the effect of gravity. Hence its vertical acceleration in the downward direction is +g, where g is the acceleration due to gravity.

Hanna Pamuła, PhD
Velocity
ft/s
Initial height
ft
Time of flight
sec
Distance
ft
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