# Horizontal Projectile Motion Calculator

Created by Hanna Pamuła, PhD candidate
Reviewed by Bogna Szyk
Last updated: Nov 24, 2021

This horizontal projectile motion calculator is a tool to solve a special case of projectile motion - the one 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 special case of general formulas. We don't need to specify the angle of launch, 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 - Vx = V, whereas Vy = 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 = – g * t² / 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:

We can combine the equations x = V * t and y = – g * t² / 2 to get rid of t. The trajectory is then equal to:

y = – g * (x / V)² / 2 = (- g * x²) / (2 * V²)

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: g * t² / 2 = h. From that equation, we can find that the time of flight equals:

t = √(2 * h / g)

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

r = V * t = v * √(2 * h / g).

We won't calculate the maximum height here, 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's 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 almost 7 seconds for the ball to reach the ground, and that the horizontal displacement is ~48.56 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.

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