MathBench

Projectile Motion Simulator

Give your projectile a push and watch it fly. Enter the force and how it is applied, the mass, the launch angle, and the site conditions — the simulator animates the flight, writes out the path equation, and reports the distance, zenith height, speeds, and time to land, with air drag and gravity you control.

Flight results

Axes are in meters and scaled independently, so the curve shows the shape of the flight, not a photo-true angle. Press Launch to replay the animation.

—launch speed
—distance flown
—zenith height
—time to land
—impact speed
—air density at site

Flight path equation

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How this calculator works

The simulator turns your push into a launch speed first: a force applied for a time transfers momentum (impulse), while a force applied over a distance — a launcher barrel, a catapult stroke — transfers energy (work). Either way it needs both numbers, because a force alone says nothing about how fast the projectile leaves; that is also why the calculator asks for a duration or a distance alongside the force.

Your site conditions set the air. Temperature and elevation feed the standard-atmosphere model to get the air density at your launch site, and the shape you pick supplies the drag coefficient — how much that silhouette resists moving through air. From there the flight is integrated step by step under gravity and drag, animated on the graph, and summarized: distance flown, zenith height (marked with a ▲), time aloft, and launch and impact speeds. The dashed curve shows where the projectile would have gone with no air at all, so the cost of drag is always visible. The equation of the ideal path is written out below the graph with your numbers plugged in.

The formula

Launch speed   v₀ = F·Δt / m            (force for a time — impulse)
               v₀ = √( 2·F·d / m )      (force over a distance — work-energy)

Ideal path     y = x·tanθ − g·x² / (2·v₀²·cos²θ)
               R = v₀²·sin 2θ / g       H = v₀²·sin²θ / (2g)      T = 2·v₀·sinθ / g

Air drag       F_d = ½·ρ·C_d·A·v²       A = π·(d/2)²   (opposes the velocity)
Air density    p = 101325·(1 − 2.25577×10⁻⁵·h)^5.25588
               ρ = p / (287.05 · T_K)

The projectile kinematics are the standard results derived in OpenStax University Physics Vol. 1 §4.3 (openstax.org); impulse and work-energy are chapters 9 and 7 of the same text. The drag equation and the preset drag coefficients come from NASA Glenn Research Center's The Drag Equation and Shape Effects on Drag pages (grc.nasa.gov) — flat plate 1.28, bullet shape 0.295, streamlined airfoil 0.045, baseball 0.30 from NASA's baseball aerodynamics work — supplemented by the classic subcritical smooth-sphere value 0.47 and cube value 1.05 from Hoerner's Fluid-Dynamic Drag, and the dimpled golf ball 0.25 from Bearman & Harvey (1976). The pressure-vs-elevation formula is the U.S. Standard Atmosphere 1976 (NOAA/NASA/USAF) troposphere law, and Moon and Mars gravity are NASA planetary fact sheet values (nssdc.gsfc.nasa.gov). With drag switched on there is no closed-form trajectory, so the simulator integrates the equations of motion with a 4th-order Runge–Kutta method at 2-millisecond steps — the graph is the actual numerical solution, not a sketch.

Worked example

Throw a baseball as hard as a good outfielder — the calculator's default. A 145 g baseball, 7.4 cm across, gets a 100 N push for 0.05 s (about what a throwing arm delivers) at 45°, at sea level, 20 °C, Earth gravity:

  1. Launch speed (impulse): v₀ = F·Δt / m = 100 × 0.05 / 0.145 = 34.48 m/s (77.14 mph)
  2. With no air, the closed form predicts: range R = v₀²·sin 90° / g = 121.3 m, height H = 30.31 m, time T = 4.973 s, and the path equation y = 1·x − 0.008247·x²
  3. Now the air: at sea level and 20 °C, ρ = 101325 / (287.05 × 293.15) = 1.204 kg/m³. Frontal area A = π(0.074/2)² = 0.004301 m², so k = ½·ρ·C_d·A = ½ × 1.204 × 0.30 × 0.004301 = 0.0007768 kg/m
  4. Integrating the flight with drag: the ball lands at 82.59 m (271 ft) after 4.441 s, reaching its zenith of 24.28 m (79.67 ft) at 2.124 s, and hits the ground at 24.58 m/s (54.98 mph) — much slower than it left, because drag has been eating energy the whole way

Air resistance cost this throw 32% of its range — that is the gap you see between the solid and dashed curves. Press Launch on the defaults and the animation reproduces every number above.

Assumptions & tips

  • The push is an idealization. A constant force for an exact time or distance is a physics-classroom launcher; a real arm or spring delivers a force that ramps up and down. If you know the actual launch speed instead, work backwards: enter it as force × time = mass × speed with any convenient pair.
  • Drag belongs to the shape, not the material. Steel and wood spheres of equal size share a drag coefficient — the material shows up as mass. That is also why a heavy ball flies farther than a light one with the same launch speed: the same drag force steals a smaller fraction of its momentum.
  • 45° is only optimal in a vacuum. With drag on, try nudging the angle down a few degrees — a baseball flies farthest near 40–42°, because drag punishes the extra hang time of a high arc more than it rewards the height.
  • Keep it subsonic. The constant-Cd model is honest up to a few hundred m/s. Near the speed of sound (~343 m/s) real drag rises sharply, so the simulator flags supersonic launches and you should read those ranges as upper bounds.
  • Changing gravity does not change the air. The atmosphere model is Earth's at every gravity setting — for the Moon or Mars, pair the gravity preset with "no air resistance" for an honest picture.
  • Spin and wind are ignored. Backspin lifts a golf ball far beyond this model (the Magnus force), and any headwind or tailwind moves the landing point. Treat results as still-air, no-spin baselines.

Frequently asked questions

Why does my projectile fly farther at high elevation or on a hot day?

Thinner air. Drag is proportional to air density, and density falls as you go up in elevation and as the air warms. The same 45-degree baseball throw that covers 82.6 m at sea level covers about 87.2 m at Denver's 1,600 m elevation — a 4.6 m gain from the air alone. That is why fly balls carry at Coors Field.

Which shape should I pick if my object is not listed?

Pick the closest silhouette as seen from the front, or choose the custom option and enter your own drag coefficient. Shape and surface texture — not material — set the drag coefficient: a steel ball and a wooden ball of the same size share Cd 0.47, and a dimpled golf ball at 0.25 beats a smooth sphere despite the rougher skin. The material still matters, but through the mass you enter, not the drag.

Why do I have to give a time or a distance along with the force?

A force alone does not determine speed — what counts is how long, or over what distance, the force acts. 100 N applied for a blink barely moves a ball; the same 100 N applied for a full second launches it ten times faster. Physics calls the two versions impulse (F times time equals momentum) and work-energy (F times distance equals kinetic energy), and the calculator supports both, with the formulas shown.

Can I simulate a throw on the Moon or Mars?

Yes — pick the gravity preset (Moon 1.62 m/s², Mars 3.71 m/s², from NASA's planetary fact sheets) or type any custom value. For the Moon, also select "no air resistance," since it has no atmosphere. The air-density model always describes Earth's atmosphere, so vacuum is the closer choice for Mars' thin CO₂ air as well.

How accurate is the drag model?

It uses the standard quadratic drag law with a constant drag coefficient, integrated numerically with a 4th-order Runge–Kutta method — the same model introductory aerodynamics courses use, good to a few percent for blunt objects at everyday speeds. It ignores spin (the Magnus force that curves a sliced golf shot), wind, and the sharp drag rise near the speed of sound (about 343 m/s), so treat supersonic answers as underestimates of the real drag.

Sources

  1. University Physics Volume 1, §4.3 — Projectile Motion — OpenStax, Rice University. openstax.orgDerives the no-drag results this page prints and draws as the dashed curve: the trajectory y = x·tanθ − g·x²/(2·v₀²cos²θ), the range v₀²·sin 2θ/g, the apex height and the time of flight. Sections 9.2 and 7.2 of the same text give the impulse and work–energy relations that convert your push into a launch speed.
  2. The Drag Equation — Beginner's Guide to Aeronautics, NASA Glenn Research Center. www1.grc.nasa.govThe quadratic drag law D = C_d · ρ · V² · A / 2 that the simulator integrates, and the definition of the reference area A taken here as the circle of the frontal width you enter.
  3. Shape Effects on Drag — Beginner's Guide to Aeronautics, NASA Glenn Research Center. www1.grc.nasa.govSource of three of the shape presets — flat plate 1.28, bullet 0.295 and streamlined airfoil 0.045 — and of the point made in the FAQ that drag coefficients belong to a silhouette measured in a wind tunnel, not to a material.
  4. Drag on a Baseball — Beginner's Guide to Aeronautics, NASA Glenn Research Center. www1.grc.nasa.govThe baseball preset, C_d ≈ 0.3, and the explanation of why a stitched ball sheds drag that a smooth ball of the same size (C_d ≈ 0.5) does not — the same effect the dimpled golf-ball preset relies on.
  5. Fluid-Dynamic Drag: Practical Information on Aerodynamic Drag and Hydrodynamic Resistance — Sighard F. Hoerner, published by the author, 1965. The classic wind-tunnel compilation behind the two presets NASA's pages do not give a single number for: the subcritical smooth sphere at C_d 0.47 and the face-on cube or box at 1.05. Cited bibliographically: the book is not published openly online.
  6. Golf Ball Aerodynamics — P. W. Bearman and J. K. Harvey, Aeronautical Quarterly, volume 27, issue 2, pages 112–122, 1976. doi.orgWind-tunnel measurements on round- and hexagonally-dimpled golf balls behind the 0.25 preset; the paper reports that dimples induce critical-Reynolds-number behaviour at a lower Reynolds number than a smooth sphere experiences, which is why the FAQ's dimpled ball beats a smooth one despite the rougher skin. Abstract is open; the full text is behind the publisher's paywall.
  7. U.S. Standard Atmosphere, 1976 (NOAA-S/T 76-1562; NASA TM-X-74335) — National Oceanic and Atmospheric Administration, National Aeronautics and Space Administration and United States Air Force, 1976. ntrs.nasa.govThe standard-atmosphere model behind the air density reported for your site: the sea-level base conditions and lapse rate the page's troposphere pressure-versus-altitude expression is reduced from, the dry-air gas constant it then divides by, and the 11,000 m top of that layer, which is where the elevation field stops.
  8. Moon Fact Sheet and Mars Fact Sheet — NASA Space Science Data Coordinated Archive, NASA Goddard Space Flight Center. Surface gravity values for the Moon (1.62 m/s²) and Mars (3.71 m/s²) offered as gravity presets. Cited bibliographically: at the time of writing the fact-sheet URLs redirect to the archive's landing page rather than to the sheets themselves.