Projectile Motion Calculator
Enter a launch speed, angle and height — then watch the trajectory fly in 3D and update live as you change the numbers. Full worked examples, charts and a physics explainer are below.
Reviewed by the ToolNestr Editorial Team — July 2026
Adjust the launch parameters — everything updates live.
Live 3D trajectory
Drag to orbit (works with touch too). The arc redraws as you change the inputs above.
Motion graphs
How projectile motion is calculated
The core idea in one line: a projectile is two independent motions happening at once — constant-speed motion sideways and constant-acceleration motion up and down — and you solve each separately, then combine them.
Everything starts by splitting the launch velocity into two components using the launch angle:
v₀ₓ = v₀ · cos(θ)
The horizontal velocity. It never changes during the flight, because nothing pushes the projectile sideways.
v₀ᵧ = v₀ · sin(θ)
The vertical velocity. Gravity steadily reduces it, stops it at the top, then speeds it up on the way down.
From there, each result is one formula. Every variable is defined right after it, so nothing is a mystery symbol:
- t = (v₀ᵧ + √(v₀ᵧ² + 2·g·h₀)) / g — the time of flight. Here g is gravity (9.81 m/s² on Earth) and h₀ is the launch height. With h₀ = 0 this simplifies to t = 2·v₀ᵧ / g.
- R = v₀ₓ · t — the range, or horizontal distance covered. Because horizontal speed is constant, range is just speed × time.
- H = h₀ + v₀ᵧ² / (2·g) — the maximum height, reached at the apex where the vertical velocity is momentarily zero.
- vᵢ = √(v₀ₓ² + (v₀ᵧ − g·t)²) — the impact speed, combining the unchanged horizontal component with the (now downward) vertical component at landing.
The complete equation set
All of the above come from the standard kinematic equations, applied separately to each axis. This is the full model the calculator runs:
Horizontal (constant velocity)
- x(t) = v₀·cosθ · t
- vₓ(t) = v₀·cosθ (constant)
- aₓ = 0
Vertical (constant acceleration −g)
- y(t) = h₀ + v₀·sinθ · t − ½g·t²
- vᵧ(t) = v₀·sinθ − g·t
- aᵧ = −g
Deriving the level-ground range formula: on flat ground (h₀ = 0) the time of flight is t = 2·v₀·sinθ / g. Substituting into R = v₀·cosθ · t and using the identity 2·sinθ·cosθ = sin(2θ) gives the compact result
R = v₀²·sin(2θ) / g
Because sin(2θ) is largest when 2θ = 90°, the range is maximised at θ = 45° — and since sin(2θ) = sin(180° − 2θ), any two angles that add up to 90° (like 30° and 60°) produce the same range. That is the mathematical reason behind the pattern you see in the table below.
Worked example 1 — level ground
Given: a ball is kicked at v₀ = 20 m/s, at θ = 30° above horizontal, from ground level (h₀ = 0), on Earth (g = 9.81 m/s²).
Worked example 2 — launched from a height (the edge case)
Given: the same 20 m/s at 30°, but now thrown from a 15 m cliff (h₀ = 15 m). This breaks the up–down symmetry, so the simple t = 2v₀ᵧ/g shortcut no longer works.
Set the initial-height slider to 15 m above to reproduce these numbers in the calculator.
Two ideas that trip students up
1. Velocity splits into two components
The red launch arrow is really the blue horizontal and green vertical arrows added together. Gravity only touches the green one — which is why the sideways speed never changes.
2. Complementary angles share a range
Five launches at the same speed. The 45° arc reaches farthest, and pairs that add to 90° (like 30° & 60°) land in exactly the same spot — one arc just flies higher and slower.
Angle vs range vs height (v₀ = 20 m/s, level ground)
Notice how range peaks at 45° while maximum height keeps climbing all the way to 90° — they are not the same thing.
| Angle | Rangem | Max heightm | Times |
|---|---|---|---|
| 15° | 20.39 | 1.37 | 1.06 |
| 30° | 35.31 | 5.10 | 2.04 |
| 45° ★ | 40.77 | 10.19 | 2.88 |
| 60° | 35.31 | 15.29 | 3.53 |
| 75° | 20.39 | 19.02 | 3.94 |
| 90° | 0.00 | 20.39 | 4.08 |
★ 45° gives the maximum range on level ground. 15° & 75° share the same range (complementary angles); so do 30° & 60°. At 90° the range is zero — the projectile goes straight up and comes straight back down.
Where projectile motion actually shows up
This is one of the most widely applied ideas in all of physics, precisely because "an object moving under gravity" describes so much of the world. Three examples where the maths on this page is doing real work:
⚽ Sport and coaching
A long-range football clearance, a basketball free throw and a javelin throw are all projectile problems. Coaches and biomechanics analysts use exactly these equations to find the release angle that maximises distance or clears a defender. The famous "45° for maximum range" only holds when the launch and landing heights match — which is why a shot-putter, releasing from about 2 m up, is actually best served by an angle a little under 42°.
🚀 Rocketry and aerospace
Model-rocket hobbyists estimate apogee (peak height) and landing distance to keep launches inside a safe field, and the same decomposition into vertical and horizontal motion is the starting point for real launch-trajectory analysis before drag, thrust and a rotating Earth are layered on top.
🎮 Game design and engineering
Every game that arcs a grenade, an arrow or an angry bird across the screen is running a projectile-motion solver behind the scenes. Civil and firefighting engineers use the same range formula to work out how far a water jet or fountain will reach at a given nozzle angle and pressure.
Common misconceptions
"Heavier objects fall faster, so they don't go as far."
In these equations mass cancels out completely — a bowling ball and a tennis ball launched identically follow the exact same arc. Mass only changes the picture once air resistance enters, which this calculator deliberately ignores. Galileo's leaning-tower demonstration was making precisely this point.
"The projectile speeds up horizontally as it falls."
Only the vertical speed changes. The horizontal component is constant from launch to landing — the object covers equal horizontal distances in equal time intervals the whole way. What increases is the total speed, because the growing downward component adds to the steady sideways one.
"45° is always the best angle for distance."
Only on level ground with no air resistance. Launch from a height and the optimum angle drops below 45°; add air resistance and it drops further still. That is why real long-drive golf shots and shot-puts are launched noticeably below 45°.
"At the very top of the arc, the object stops."
Only the vertical velocity is zero at the apex. The object is still moving horizontally at full speed — which is exactly why the top of a trajectory is a smooth curve, not a sharp point where everything pauses.
Assumptions — and where this model breaks down
This calculator uses the ideal (drag-free) projectile model, the same one taught in introductory physics. It is exact in a vacuum and an excellent approximation for dense, slow, compact objects over short distances — but it is a model, and it is honest to know its limits:
Air resistance is ignored
Real drag grows with the square of speed and can cut the range of a fast, light object (a ping-pong ball, a golf ball) by anywhere from 10% to over 50%. The heavier and slower the object, the closer reality gets to these numbers.
No spin / Magnus effect
A spinning ball curves because of the Magnus force (think of a football "banana" kick or a curveball). This model assumes no spin, so the path stays in a single vertical plane.
Gravity is constant and uniform
g is treated as a fixed value for the whole flight. That is fine near the surface, but not for very high or very long-range trajectories where g changes with altitude and the Earth curves away beneath the object.
No Coriolis or wind
Earth's rotation (the Coriolis effect) and wind both nudge real long-range projectiles — which is why artillery and ballistics use far more complex models. For classroom-scale problems their effect is negligible.
Formula sources & further reading
The equations used here are the standard projectile-motion kinematics found in every introductory physics curriculum. They are traceable to these widely used references:
- • OpenStax, University Physics Volume 1 — §4.3 "Projectile Motion" (free, peer-reviewed). openstax.org
- • Halliday, Resnick & Walker, Fundamentals of Physics (11th ed.) — Chapter 4, Motion in Two and Three Dimensions.
- • Serway & Jewett, Physics for Scientists and Engineers — Chapter 4, Motion in Two Dimensions.
- • Young & Freedman, University Physics — Chapter 3, Motion in Two or Three Dimensions.
Standard value of gravitational acceleration used: g = 9.80665 m/s² (rounded to 9.81), per the CGPM standard-gravity definition. Results are rounded to two decimal places.
How to use this calculator
Set the launch
Drag the speed, angle and height sliders, or type exact values. Switch between metric and imperial with the toggle.
Read the results
Range, max height, time of flight and impact speed update instantly, and the 3D arc and charts redraw to match.
Explore other worlds
Tap Moon, Mars or Jupiter to change gravity and watch how far the same launch would travel elsewhere.
Related tools
Frequently asked questions
What is projectile motion?
Projectile motion describes the trajectory of an object launched into the air, subject only to gravity (neglecting air resistance). The path is a parabola: horizontal motion at constant speed, vertical motion with constant downward acceleration g.
What is the optimum angle for maximum range?
On level ground with no air resistance, 45° gives the maximum range, because 45° splits the launch velocity equally between horizontal and vertical components. If you launch from a height, the best angle drops below 45°.
How do I calculate maximum height?
Maximum height is reached when the vertical velocity becomes zero. The formula is H = h₀ + (v₀·sinθ)² / (2g), where v₀ is launch speed, θ is the launch angle, h₀ is the starting height and g is gravity.
Why does the horizontal velocity stay constant?
With air resistance neglected, no horizontal force acts on the projectile (gravity acts vertically). Newton's first law then says the horizontal velocity component stays constant for the whole flight.
Do heavier objects fall faster in these calculations?
No. In a vacuum, mass cancels out of the equations of motion, so a cannonball and a marble launched identically follow the exact same path. Mass only matters once air resistance is included, which this calculator ignores.
How do I calculate range on uneven ground?
When the landing point sits at a different height than the launch point, solve the vertical equation h₀ + v₀·sinθ·t − ½gt² = 0 for the time of flight t, then multiply t by the horizontal velocity component to get the range. This calculator does exactly that when you set an initial height.