Projectile Motion Calculator

Background.

This projectile motion calculator solves the canonical no-drag two-dimensional kinematics problem — what trajectory does an object follow when it is launched with some initial speed at some angle above the horizontal and then left alone in a uniform gravitational field — and returns the four numbers students, engineers, ballistics analysts, and sports coaches actually ask for: horizontal range, time of flight, maximum height, and the two velocity components at launch. Type in the initial speed V₀ in metres per second, the launch angle θ in degrees above horizontal, the height of the launch point above the landing surface, and the local gravitational acceleration (9.81 m/s² for Earth, 1.62 for the Moon, 3.71 for Mars), and the calculator returns the four trajectory quantities in SI units.

The physics underneath the widget is the great clean insight that Galileo published in 1638 in the 'Two New Sciences' and that has been the cornerstone of mechanics ever since: motion in two dimensions under constant gravity is just the superposition of two independent one-dimensional motions. The horizontal motion is uniform — constant velocity V_x = V₀·cos(θ), no acceleration, no force — because gravity has no horizontal component. The vertical motion is uniformly accelerated free fall — initial velocity V_y = V₀·sin(θ), constant downward acceleration g — exactly the same as if you had dropped or thrown the object straight up. The two motions are decoupled, which means you can solve them separately and recombine.

Solve the vertical equation y(t) = h₀ + V_y·t − ½·g·t² for the moment y reaches the landing surface (y = 0), substitute that flight time into the horizontal equation x = V_x·t, and you have the range. The time of flight from a ground-level launch is t = 2·V₀·sin(θ)/g; the maximum height above the launch point is V₀²·sin²(θ)/(2g); the range over level ground is R = V₀²·sin(2θ)/g. That last identity — sin(2θ) is symmetric about θ = 45° and peaks there — is the famous result that a 45° launch from ground level gives the maximum horizontal range, with complementary angles (e.g. 30° and 60°) producing identical ranges.

When the launch point and the landing surface are not at the same elevation — a cannon firing off a cliff, a basketball released 2.1 m above the floor and falling into a rim 3.05 m up, a golf ball driven from a tee box onto a fairway 5 m below — the symmetry breaks and the optimum angle drops below 45°. The calculator handles both cases without you having to remember which formula to use: the vertical equation is solved with the full quadratic for arbitrary initial height, so the answer is correct whether h₀ is zero or 50 m.

Below the widget you will find a step-by-step derivation of every formula, an honest discussion of when the no-drag assumption is fine (heavy dense objects at modest speeds — a thrown rock, a shot put, a cannonball under 300 m/s) and when it is dangerously wrong (anything that depends on its trajectory for actual use — bullets, javelins above 25 m/s, golf balls in flight, tennis serves), worked examples for cannon-fire, basketball, and javelin, and the standard correction approaches when air resistance does matter (the quadratic drag model F_drag = ½·ρ·C_d·A·v², ballistic coefficients, and the Lambert W-function solution for the linear-drag special case).

The formula identifier is projectileMotion.calculate and the engine returns all five outputs as a single self-consistent set. Air resistance, wind, Coriolis deflection, Magnus force on a spinning ball, and the variation of g with altitude are all set to zero in this model — for the great majority of textbook problems and back-of-the-envelope sports questions, those simplifications are appropriate and the answer this calculator returns matches the closed-form physics-textbook solution to better than one part in a million.

What is projectile motion calculator?

Projectile motion is the trajectory of an object launched into the air and acted on only by gravity (and, in more complete models, air resistance). The phrase usually refers to the idealized no-drag case in which the only force is the constant downward gravitational acceleration g, and the object is treated as a point mass with no spin, no lift, and no wind acting on it. Under those assumptions the trajectory is a parabola — Galileo's discovery — and the entire two-dimensional problem decomposes into two independent one-dimensional problems: uniform motion at constant velocity V_x = V₀·cos(θ) along the horizontal axis, and uniformly accelerated motion (V_y = V₀·sin(θ) initially, accelerating downward at g) along the vertical axis. The four quantities a projectile-motion problem typically asks for are the horizontal range R (where does it land?), the time of flight T (how long is it in the air?), the maximum height H (how high does it go?), and the velocity components at any moment (how fast is it moving horizontally and vertically?). All four follow from the two decoupled equations of motion x(t) = V_x·t and y(t) = h₀ + V_y·t − ½·g·t² with appropriate boundary conditions. The model applies cleanly to dense objects at modest speeds over short distances, where the air-drag force is small compared to gravity. It applies poorly — sometimes dangerously poorly — to light objects (badminton shuttlecocks, ping-pong balls), spinning objects (curveballs, golf balls), high-speed objects (bullets, artillery shells beyond about 300 m/s), and any object whose flight is long enough for air drag to integrate into a meaningful correction. For those cases you need a numerical integrator with a drag model — quadratic drag at high Reynolds numbers, linear drag at very low ones — and the closed-form parabolic answer is only a first approximation.

How to use this calculator.

1. 01Enter the initial velocity V₀ in metres per second. For a bullet leaving a rifle muzzle this is hundreds of m/s; for a thrown baseball it's about 40 m/s; for a basketball jump shot it's about 7 m/s. Convert from km/h by dividing by 3.6, and from mph by dividing by 2.237.
2. 02Enter the launch angle θ in degrees above the horizontal. 0° is a perfectly flat throw; 45° gives the maximum range from level ground; 90° is straight up (range = 0). Use any value 0° ≤ θ ≤ 90°.
3. 03Enter the initial height h₀ in metres — the height of the launch point above the landing surface. Use 0 for a ground-to-ground shot, the release height for a basketball shot, the cliff height for an artillery problem, or the tee elevation minus the fairway elevation for a golf drive.
4. 04Set the gravitational acceleration g. Default is 9.81 m/s² (Earth standard, sometimes quoted as 9.80665 m/s² for technical applications). Use 1.62 for the Moon, 3.71 for Mars, 24.79 for Jupiter, 8.87 for Venus.
5. 05Read the primary result — horizontal range — and the four supporting values: time of flight, maximum height above the landing surface, and the horizontal and vertical components of the initial velocity vector.
6. 06For maximum range on level ground at fixed V₀, try θ = 45°. For maximum range from a cliff (h₀ > 0), the optimum angle is less than 45° — sweep θ from 30° to 45° in the calculator and look for the peak.

The formula.

The model takes the launch point as the origin (or as (0, h₀) if there is an initial height), the horizontal direction as x with positive x in the direction of motion, and the vertical direction as y with positive y upward. The launch velocity vector has magnitude V₀ at angle θ above the horizontal, so its components are:

V_x = V₀ · cos(θ) (horizontal component, constant for all time in the no-drag model) V_y = V₀ · sin(θ) (initial vertical component)

The equations of motion follow directly from Newton's second law with the only force being gravity:

x(t) = V_x · t (horizontal position, uniform motion) y(t) = h₀ + V_y · t − ½ · g · t² (vertical position, uniformly accelerated) v_x(t) = V_x (horizontal velocity, constant) v_y(t) = V_y − g · t (vertical velocity, decreasing linearly)

The time of flight T is the moment when y(T) = 0 (the projectile reaches the landing surface). Solving the quadratic h₀ + V_y·T − ½·g·T² = 0 by the quadratic formula and taking the positive root:

T = (V_y + √(V_y² + 2·g·h₀)) / g

When h₀ = 0 this collapses to the familiar T = 2·V_y/g = 2·V₀·sin(θ)/g for a level-ground shot. The horizontal range is then simply R = V_x · T:

R = (V₀·cos(θ)) · (V₀·sin(θ) + √((V₀·sin(θ))² + 2·g·h₀)) / g

Again, with h₀ = 0 this reduces via the double-angle identity 2·sin(θ)·cos(θ) = sin(2θ) to the textbook range formula:

R = V₀² · sin(2θ) / g (level-ground range)

The sin(2θ) factor peaks at 2θ = 90°, i.e. θ = 45°, which is why a 45° launch maximises range from ground level. Complementary angles produce equal ranges: sin(2 · 30°) = sin(60°) = sin(120°) = sin(2 · 60°). Maximum height H is reached when v_y(t) = 0, at time t_peak = V_y/g:

H = h₀ + V_y² / (2·g) = h₀ + V₀² · sin²(θ) / (2·g)

The calculator implements these closed-form expressions exactly. Internally it converts θ from degrees to radians, computes V_x and V_y, computes T from the full quadratic so that arbitrary initial heights work correctly, then derives R and H. All five outputs are returned as a self-consistent set: substituting them back into x(T) and y(T) recovers (R, 0) to floating-point precision.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Galilei, G. (1638). Discorsi e dimostrazioni matematiche, intorno a due nuove scienze (Discourses and Mathematical Demonstrations Relating to Two New Sciences). Day Four — On the Motion of Projectiles. Leiden: Elsevier. The original derivation of the parabolic trajectory, including Galileo's decomposition of motion into independent horizontal and vertical components.
2. [2]Halliday, D., Resnick, R. & Walker, J. (2014). Fundamentals of Physics, 10th ed., Chapter 4 'Motion in Two and Three Dimensions', Sections 4-5 to 4-7 on projectile motion. Wiley. The standard university-level treatment of projectile-motion kinematics, including derivation of range, height, and time-of-flight formulae and worked examples.
3. [3]NASA Glenn Research Center — Beginners' Guide to Aeronautics, Ballistic Flight Equations. Derivation of the no-drag trajectory equations U = U₀, V = V₀ − g·t, x = U₀·t, y = V₀·t − ½·g·t², together with the time-to-apogee and maximum-altitude results. Maintained by NASA's K-12 educational programme.
4. [4]Feynman, R. P., Leighton, R. B. & Sands, M. (1963). The Feynman Lectures on Physics, Volume I, Chapter 8 'Motion' and Chapter 9 'Newton's Laws of Dynamics', covering the independence of horizontal and vertical motion under gravity and Feynman's discussion of why mass cancels out of free-fall trajectories.
5. [5]Linthorne, N. P. (2001). Optimum release angle in the shot put. Journal of Sports Sciences, 19(5), 359–372. Peer-reviewed biomechanics paper deriving the optimum release angle for the shot put as a function of release height and release speed, and explaining why elite athletes throw below the 45° ballistic optimum.
6. [6]Bartonietz, K. & Borgström, A. (1995). The throwing events at the World Championships in Athletics 1995, Göteborg — Technique of the world's best athletes. Part 2: Discus and javelin throw. New Studies in Athletics, 10(4), 19–44. Empirical measurement of release angles, speeds, and heights for world-class javelin throwers, finding optimum release angles of 32–36°.
7. [7]Fontanella, J. J. (2006). The Physics of Basketball. Johns Hopkins University Press. Quantitative treatment of basketball trajectories, including the calculation of the optimum free-throw release angle (~51°) and the effective rim diameter as a function of entry angle.
8. [8]Parker, G. W. (1977). Projectile motion with air resistance quadratic in the speed. American Journal of Physics, 45(7), 606–610. Peer-reviewed pedagogical paper on the quadratic-drag projectile problem, its lack of a closed-form solution, and standard numerical integration approaches.