Horizontal Projectile Motion Calculator
Enter the launch height and horizontal launch speed to find the time of flight, horizontal range, final vertical speed, resultant impact speed, and impact angle. Switch freely between metric and imperial units. The trajectory chart updates live so you can see the full parabolic path from launch to impact. Air resistance is neglected, making this the standard model used in introductory physics courses.
Formula
Worked example
A ball rolls off a table 20 m high at 15 m/s. Time of flight: t = sqrt(2 x 20 / 9.81) = 2.019 s. Horizontal range: x = 15 x 2.019 = 30.29 m. Final vertical speed: vy = 9.81 x 2.019 = 19.81 m/s downward. Impact speed: vf = sqrt(15^2 + 19.81^2) = 24.85 m/s. Impact angle: atan(19.81 / 15) = 52.9 deg below horizontal.
What is horizontal projectile motion?
Horizontal projectile motion is the special case where an object is launched with a purely horizontal velocity from an elevated point, and gravity then curves its path downward. Because the initial vertical velocity is zero, the downward acceleration builds from zero during the entire flight. Horizontally, the object moves at constant speed (no air resistance), so the horizontal and vertical motions are completely independent. This gives a parabolic trajectory that is easy to analyze with two separate equations: one for the constant horizontal motion and one for free fall. Classic examples include a ball rolling off a table, a projectile fired horizontally from a cliff, a stone thrown horizontally from a bridge, and a ball launched from the end of a ramp.
The four key equations
Time of flight depends only on the launch height and gravity: t = sqrt(2h/g). Horizontal range is the launch speed multiplied by that time: x = v x t. The vertical speed at impact is g x t, because the object starts with zero vertical velocity and accelerates at g the whole way down. The resultant impact speed combines horizontal and vertical components using the Pythagorean theorem: vf = sqrt(v^2 + vy^2). The impact angle below the horizontal follows from the inverse tangent: theta = atan(vy / v). Notice that the time of flight does not depend on the horizontal speed at all: two objects launched horizontally from the same height will land at the same time regardless of their speeds, a fact that can be confirmed with a side-by-side drop experiment.
Using this calculator for physics problems
Select your unit system first so every input and output shows the right label. Enter the launch height (how far above the ground the launch point sits) and the horizontal speed. If you are working on another planet or the Moon, change the gravity setting to match. The show-your-work panel under the results walks through each formula step with your exact numbers, which is useful for checking homework or understanding where each answer comes from. The trajectory chart shows the parabolic path from launch point to impact point, with height on the vertical axis and horizontal distance on the horizontal axis.
Assumptions and limitations
This calculator assumes a flat, level landing surface at the same height as the base of the launch point (height = 0). Air resistance is completely neglected, which is a good approximation at low speeds and short ranges but increasingly inaccurate for fast or dense objects over long distances. The calculation also assumes a uniform gravitational field, which is valid for any trajectory well within the atmosphere. The Earth gravity value used (9.81 m/s^2) is the internationally adopted standard; local variations due to altitude or latitude are not accounted for.
Gravity on different celestial bodies
| Body | Gravity (m/s^2) | Relative to Earth |
|---|---|---|
| Moon | 1.62 | 0.165x |
| Mars | 3.72 | 0.379x |
| Earth | 9.81 | 1.000x |
| Jupiter | 24.79 | 2.528x |
Standard gravitational acceleration values used in projectile motion problems. Earth value is the internationally adopted standard.
Frequently asked questions
Does the horizontal speed affect how long the projectile is in the air?
No. Time of flight depends only on the launch height and gravity (t = sqrt(2h/g)). A ball launched at 5 m/s and one launched at 50 m/s from the same height will both hit the ground at exactly the same time. The faster ball simply lands much farther away.
What is the launch angle in horizontal projectile motion?
The launch angle is always zero degrees (horizontal). That is what makes this the "horizontal" case. The impact angle, however, is not zero: gravity adds a downward component during the flight, so the projectile hits the ground at a downward angle. That impact angle depends on both the horizontal speed and the final vertical speed.
Why is the trajectory a parabola?
Horizontal displacement grows linearly with time (x = v x t), while vertical displacement grows as the square of time (y = h - g*t^2 / 2). Eliminating t gives y as a quadratic function of x, which is exactly the equation of a downward-opening parabola.
How does gravity on the Moon change the result?
The Moon's gravitational acceleration is about 1.62 m/s^2, roughly one-sixth of Earth's. Because time of flight scales as sqrt(1/g) and range scales as 1/sqrt(g), a projectile on the Moon takes about sqrt(6) = 2.45 times longer to fall and lands about 2.45 times farther than on Earth, given the same height and launch speed.
How is horizontal projectile motion different from general projectile motion?
In general projectile motion the launch angle can be any value from 0 to 90 degrees, so the object starts with both a horizontal and a vertical velocity component. In horizontal projectile motion the launch angle is exactly zero, which means the vertical velocity starts at zero and the object is essentially in free fall from the moment of launch. This simplifies the equations considerably and is the version most commonly used in introductory physics courses.
What units can I use?
Choose Metric (metres and metres per second) or Imperial (feet and feet per second) from the unit system selector. Time of flight is always shown in seconds regardless of the unit system. All other inputs and outputs switch together when you change the system.