About the Projectile Motion Calculator
The Projectile Motion Calculator solves the classic 2D kinematic problem: given an initial velocity, launch angle, and launch height, it returns the time of flight, maximum height, total range, and the parabolic trajectory rendered as a real-time chart. It supports custom gravitational constants so you can model launches on Mars, the Moon, or any user-supplied value — useful for physics classrooms and orbital-curious students.
It is built for high-school and introductory undergraduate physics students working through chapter problems, science teachers building lecture demonstrations, model-rocketry hobbyists doing rough motor-impulse checks, and game developers sanity-checking projectile arcs. The 45-degree-maximum-range result is reproduced and explained alongside the why — that it only holds for level ground and no air resistance, which is rarely true.
All physics math runs in JavaScript inside your browser. No velocity, angle, or scenario input ever leaves your device — the page makes no network call after first load. The trajectory chart is rendered locally with Canvas; the data behind it never touches a server. URLs containing scenario parameters encode them client-side and do not phone home.
The model assumes a vacuum: no air resistance, no spin-induced lift or drag, no wind. For artillery-grade accuracy, model-rocket altitude prediction, or any real-world ballistic problem, you need numerical integration with a drag coefficient (Cd) and air density at the launch altitude. Use the output as the ideal-case upper bound, then expect real-world ranges to land roughly twenty to forty percent shorter once drag is honored.
Projectile Motion Equations Explained
Projectile motion describes the path of any object launched into the air and subject only to gravity (ignoring air resistance in the ideal case). The motion is two-dimensional: horizontal motion at constant velocity Vx = V₀ cos(θ), and vertical motion under uniform gravitational acceleration. The initial vertical component Vy0 = V₀ sin(θ) determines how high and how long the projectile stays aloft. These two independent components combine to trace a parabolic arc — the same curve Galileo first described in the early 17th century.
Key Formulas
The time of flight when launching from height h above ground is found by solving the quadratic y(t) = h + Vy0 t − ½g t² = 0, yielding t = (Vy0 + √(Vy0² + 2gh)) / g. The range is simply Vx × t. Maximum height equals h + Vy0² / (2g) when the vertical component momentarily reaches zero. Impact velocity combines the unchanged horizontal speed with the final vertical speed: Vimpact = √(Vx² + Vy,impact²).
Why 45° Gives Maximum Range (and When It Doesn’t)
For a projectile launched from ground level in a vacuum, 45° maximizes range because it perfectly balances horizontal distance per unit time with time in the air. The range formula R = V₀² sin(2θ) / g peaks when sin(2θ) = 1, i.e. θ = 45°. However, launching from an elevated platform shifts the optimum angle below 45° because the projectile has extra falling time regardless of angle. Similarly, complementary angles (like 30° and 60°) produce identical ranges from ground level — a useful symmetry for physics problem-solving. In real-world scenarios with air resistance, the optimal angle depends on the projectile’s mass, cross-sectional area, and drag coefficient.
How Gravity Differs Across Planets
Gravity determines how quickly a projectile decelerates vertically. On Earth, g ≈ 9.81 m/s². On the Moon, with just 1.62 m/s², a projectile launched at 50 m/s and 45° travels about six times farther than on Earth — which is why lunar golf drives are legendary. Mars sits between the two at 3.72 m/s², while Jupiter’s crushing 24.79 m/s² means projectiles barely leave the ground. These differences are critical for planning everything from spacecraft landings to hypothetical sports on other worlds.
Real-World Projectile Motion: Air Resistance Effects
The idealized parabolic trajectory assumes no air resistance, but real projectiles face drag force proportional to the square of their velocity: Fdrag = ½ ρ Cd A v². This force acts opposite to the velocity vector at every instant, reducing both height and range. The effect is dramatic for lightweight, fast objects — a golf ball hit at 70 m/s loses roughly 40% of its vacuum range to drag. Spin adds further complexity through the Magnus effect, curving the trajectory sideways or altering lift. Our premium air resistance mode uses numerical integration (Euler’s method) to compute realistic trajectories with user-specified drag coefficient, mass, and cross-sectional area.
Projectile Motion Practice Problems
Problem 1: A ball is thrown at 20 m/s at 30° from the top of a
15 m building. How far from the base does it land?
Try it: Enter v = 20, θ = 30°, h = 15 m, Earth gravity.
The calculator gives the range instantly — check against your hand calculation using the
quadratic formula for time of flight.
Problem 2: A cannon fires a shell at 200 m/s. At what two angles
will it hit a target 3 km away?
Try it: Use the premium “Hit the Target” mode with v = 200 m/s
and target = 3000 m. The tool finds both the low-angle and high-angle solutions
— demonstrating the complementary angle symmetry of ground-level launches.
Problem 3: Compare a soccer kick at 25 m/s and 45° on Earth versus
Mars. How much farther does it travel on Mars?
Try it: Run the calculation on Earth, note the range, then switch to Mars. The planet
comparison callout shows the exact percentage difference.
Looking for related tools? Our Graphing Calculator can plot custom equations including parametric trajectories, and the Statistics Calculator helps analyze experimental launch data. Explore all Math & Science tools.
Frequently Asked Questions
What is the formula for projectile range?
For launch and landing at the same height, range R = V0^2 * sin(2*theta) / g. When launched from height h, range is V0*cos(theta) * (V0*sin(theta) + sqrt((V0*sin(theta))^2 + 2*g*h)) / g.
What launch angle gives maximum range?
From ground level with no air drag, 45 degrees maximizes range. With a nonzero launch height, the optimal angle is slightly less than 45 degrees. With air drag, real-world optimum is often 35 to 42 degrees.
What is the formula for maximum height?
Maximum height above the launch point equals (V0*sin(theta))^2 / (2*g). Adding the launch height h gives the absolute peak altitude.
What value of g should be used?
Earth's standard gravity is 9.81 m/s2 (32.2 ft/s2). Other useful values: Moon 1.62 m/s2, Mars 3.71 m/s2, Jupiter 24.79 m/s2. These assume ideal free-fall with no atmospheric drag.
Does this calculator include air resistance?
No. It uses the classical ideal-projectile model with gravity only. Air drag would reduce range and maximum height and shift the optimum launch angle below 45 degrees.