Horizontal Projectile Motion Calculator finds the range, time of flight and impact speed of a level launch from a height. It also solves backward for the launch velocity or height.
Pick the two values your problem gives. The launch is level, with no upward or downward push.
Check Point and Units
Step-by-Step Working
- Known values in SI units: v = 5 m/s, h = 1.25 m, g = 9.80665 m/s²
- Time of flight: t = √(2h ÷ g) = √(2 × 1.25 ÷ 9.80665) = 0.5049 s
- Range: R = v × t = 5 × 0.5049 = 2.52 m
- Vertical speed at impact: g × t = 9.80665 × 0.5049 = 4.95 m/s
- Impact speed: √(v² + (gt)²) = √(5² + 4.95²) = 7.04 m/s at 44.72° below level
- Result: it lands 2.52 m out after 0.5049 s, launched at 5 m/s from 1.25 m
Trajectory Table
| Time | Distance Out | Height | Speed |
|---|---|---|---|
| 0 s | 0 m | 1.25 m | 5 m/s |
| 0.101 s | 0.5049 m | 1.2 m | 5.1 m/s |
| 0.202 s | 1.01 m | 1.05 m | 5.38 m/s |
| 0.3029 s | 1.51 m | 0.8 m | 5.82 m/s |
| 0.4039 s | 2.02 m | 0.45 m | 6.38 m/s |
| 0.5049 s | 2.52 m | 0 m | 7.04 m/s |
Trajectory Graph
Horizontal Projectile Motion Calculator Results
Horizontal projectile motion is the path of an object launched level from a height, with no upward or downward push. A ball rolling off a table is the standard case. The Horizontal Projectile Motion Calculator solves it from any two values the problem gives.
The Known Values menu has five pairs. Velocity and height is the usual one, and it returns the range and the time of flight. The other pairs work backward, so a measured range and height give the launch velocity, and a velocity and range give the height.
Four cards follow the headline. They show the time of flight, the impact speed with its angle below level, the height at any check distance from the launch edge, and the same results in feet. With the default 5 m/s from 1.25 m, the object lands 2.52 m out after 0.5049 s.
Every answer comes with the working written out line by line, so the tool also serves as a horizontal projectile motion calculator with steps. A trajectory table and a graph of the half parabola sit below the working.
Horizontal Projectile Motion Equations
The motion splits into two independent parts, as OpenStax University Physics sets out for all projectiles. Horizontally there is no acceleration, so the speed stays at the launch value v. Vertically the object falls from rest under gravity g.
$$x = v t$$
$$y = h – \frac{1}{2} g t^2$$
These are the two position equations the Horizontal Projectile Motion Calculator starts from, with h as the launch height. Setting y to zero gives the time of flight. It depends only on the height and gravity, not on the launch speed.
$$t = \sqrt{\frac{2h}{g}}$$
The horizontal distance covered in that time is the range R. This is the formula for horizontal distance that most problems ask for, and it is the third of the three formulas for a level launch, after the two position equations.
$$R = v \sqrt{\frac{2h}{g}}$$
Removing time from the two position equations gives the shape of the path. The height falls with the square of the distance out, which makes the trajectory half of a parabola. The check point card uses this line.
$$y = h – \frac{g x^2}{2 v^2}$$
At landing the horizontal speed is still v, and the vertical speed has grown to gt. The impact speed combines the two, and the impact angle below level comes from their ratio, which is 44.72° for the default launch.
$$v_{impact} = \sqrt{v^2 + 2 g h}$$
Horizontal Range by Height and Speed
The table shows how to calculate the horizontal range of a projectile across common heights on Earth. The time of flight is the same in every column of a row. The range simply scales with the launch speed.
| Launch height | Time of flight | Range at 5 m/s | Range at 10 m/s | Range at 20 m/s | Vertical speed at impact |
|---|---|---|---|---|---|
| 1 m | 0.452 s | 2.26 m | 4.52 m | 9.03 m | 4.43 m/s |
| 2 m | 0.639 s | 3.19 m | 6.39 m | 12.77 m | 6.26 m/s |
| 5 m | 1.01 s | 5.05 m | 10.1 m | 20.2 m | 9.9 m/s |
| 10 m | 1.428 s | 7.14 m | 14.28 m | 28.56 m | 14 m/s |
| 20 m | 2.02 s | 10.1 m | 20.2 m | 40.39 m | 19.81 m/s |
| 50 m | 3.193 s | 15.97 m | 31.93 m | 63.87 m | 31.32 m/s |
Doubling the speed doubles the range. Doubling the height does not, because the time of flight grows only with the square root of the height. Going from 5 m to 20 m, four times the height, just doubles the time and the range.
Gravity changes the answer the same way. The default launch of 5 m/s from 1.25 m carries 2.52 m on Earth, 4.11 m on Mars and 6.25 m on the Moon. The gravity menu in the Horizontal Projectile Motion Calculator holds those presets.
Horizontal Projectile Motion Calculator With Solution
The five rows below run one problem through each setting of the Known Values menu. The second and third are exercises from the OpenStax chapter, a bullet fired level from shoulder height and a crate dropped from a plane in level flight.
| Problem | Known values | Answer | Also returned |
|---|---|---|---|
| Ball rolled off a table | v = 5 m/s, h = 1.25 m | Range 2.52 m | Time 0.5049 s, impact 7.04 m/s at 44.72° |
| Bullet from shoulder height | v = 200 m/s, h = 1.5 m | Range 110.62 m | Time 0.5531 s |
| Crate dropped from a plane | v = 500 km/h, h = 800 m | Range 1,774.1 m | Time 12.77 s, impact 187.03 m/s |
| Marble that lands 3 m out | h = 1 m, R = 3 m | Launch velocity 6.64 m/s | Time 0.4516 s |
| Jump that must clear 8 m | v = 10 m/s, R = 8 m | Launch height 3.14 m | Time 0.8 s |
The first row is the textbook answer of 0.5 s and 2.5 m, which comes from rounding g to 9.8 or 10. The Horizontal Projectile Motion Calculator uses standard gravity of 9.80665 m/s² unless another value is chosen. The crate keeps the plane’s forward speed, so it lands 1.77 km ahead of the release point.
The bullet row shows the limit of the model. With no air, a bullet at 200 m/s lands after 0.55 s like any other object. A real bullet slows from drag on the way, which is what the Ballistic Coefficient Calculator describes, and a falling object with drag is covered by the Free Fall with Air Resistance Calculator.
Vertical and Horizontal Motion Together
The fall takes exactly as long as a straight drop from the same height. A ball rolled off a 1.25 m table and a ball dropped beside it land together after 0.5049 s. The vertical half of the problem is plain free fall, the case the Free Fall Calculator handles alone.
The horizontal half is uniform motion. Nothing pushes or slows the object sideways, so the horizontal velocity at landing equals the launch velocity. That makes the tool a horizontal velocity calculator too, since the Height and Range setting returns it from a measured landing spot.
The path is not a straight slope. At half the range, only a quarter of the height is gone, so the default ball is still 0.9375 m up when it is 1.26 m out. The check point card in the Horizontal Projectile Motion Calculator gives the height at any other distance, such as 1.05 m high at 1 m out.
A launch that is tilted up or down is a different problem, because the object then starts with vertical speed. For a horizontal projectile motion calculator with angle, the Car Jump Distance Calculator takes a launch speed, a ramp angle and the heights at both ends.
Projectile Motion Questions
What is the horizontal distance of a projectile called?
It is called the range, and it is the headline result of the Horizontal Projectile Motion Calculator. For a level launch the horizontal range of a projectile is the launch speed times the time of flight. There is no maximum for a level launch, since more speed or more height always carries it farther.
The familiar maximum belongs to launches from the ground at an angle. There the range is greatest at 45°, where it equals v² ÷ g. A 20 m/s launch at 45° reaches 40.79 m on level ground.
What are the two types of projectile motion?
Horizontal projectile motion starts level from a height, and angled or oblique projectile motion starts at an angle above or below level. A vertical throw straight up or down is the limiting case with no horizontal speed, and it is ordinary one-dimensional motion.
All of them obey the same rule. Gravity acts only on the vertical part, and the horizontal part keeps a constant speed. A vertical projectile motion calculator needs only the free fall equations, while the angled case needs both parts.
What are 5 examples of projectile motion?
A ball rolling off a table and a package dropped from a plane in level flight are both horizontal launches. A kicked football, a basketball shot and water from a tilted hose are angled launches. In each one the only force after release is gravity, if air resistance is ignored.
What are the three equations of motion?
They are v = u + at, s = ut + ½at² and v² = u² + 2as, for motion with constant acceleration. Projectile problems apply them twice, once sideways with a = 0 and once vertically with a = g. The Displacement Calculator works with these three directly.
They are not the same as Newton’s laws of motion. The third law of motion says forces come in equal and opposite pairs, written FAB = −FBA. Lists of the types of motion also vary by textbook, and projectile motion counts as curvilinear motion in all of them.