Projectile Motion Calculator

Free Online Interactive 30+ Presets

Free projectile motion calculator with trajectory visualization. Calculate range, time of flight, maximum height with or without air resistance. 30+ presets for basketball, golf, baseball, Moon/Mars physics.

Controls
Pre-configured challenges with targets
Compare shots: Enable Keep Shots to overlay trajectories. Turn on Overlay in the toolbar to see drag vs no-drag. Use the Scrubber to replay any shot frame-by-frame. Drag from the cannon to set v and θ directly.
Live State
t
–
x
–
y
–
vx
–
vy
–
|v|
–
θv
–
Results
Time of Flight
–
Total airborne time, from launch to ground impact.
Range
–
Horizontal distance. Maximized at 45° without drag.
Max Height
–
Highest point, where vy = 0 momentarily.
Final Speed
–
Speed at impact. Without drag, equals launch speed when y₀ = 0 (energy conservation).
Range
R = v₀² sin(2θ) / g
Time of Flight
T = 2v₀ sin(θ) / g
Max Height
H = v₀² sin²(θ) / (2g)
Drag Force
Fd = ½ρCdAv²
With drag, no closed-form exists — solved via Euler integration (dt = 0.005 s).
  • 45° rule: Without drag and y₀ = 0, 45° maximizes range. With a starting height, the optimal angle is slightly lower.
  • Complementary angles: 30° and 60° give the same range (without drag). Try it!
  • Drag shifts the optimal angle below 45° — the faster you go, the more drag matters.
  • Mass only matters with drag. In vacuum, a feather and cannonball follow the same path.
  • Altitude reduces air density (ρ), so there’s less drag at high elevation (e.g., Everest Summit scenario).
  • Energy conservation: Without drag, KE + PE stays constant. Drag dissipates energy as heat.
1. Find the Optimal Angle
Goal: Discover which angle gives maximum range.
  1. Set speed to 25 m/s, drag OFF, Earth gravity.
  2. Launch at 30°, 45°, and 60° (keep shots on).
  3. Compare the three ranges. Which is longest?
  4. Now turn drag ON (Tennis preset). Repeat. Does the optimal angle change?
2. Earth vs Moon
Goal: See how gravity affects trajectory.
  1. Launch at 45°, 25 m/s on Earth. Note the range.
  2. Switch gravity to Moon (1.62 m/s²). Launch again.
  3. Compare: the Moon shot goes ~6× farther!
3. Feather vs Cannonball
Goal: Understand why mass matters with drag.
  1. Turn drag ON. Select the Feather preset. Launch at 45°.
  2. Reset shots. Select Cannonball. Launch at same angle & speed.
  3. The cannonball goes much farther — its high mass resists drag.
4. Energy Trade-off
Goal: Watch KE and PE exchange in real time.
  1. Click the Energy chart tab below the canvas.
  2. Launch at 60° with drag OFF. Watch KE dip at the apex as PE peaks.
  3. Now turn drag ON and launch again. Notice energy is lost to drag.
Trajectory
How to use: Axes show meters. Solid line = current model; dashed = no-drag overlay (if enabled). Drag the target or cannon to interact!
Charts

Frequently Asked Questions

Enter the initial velocity (m/s), launch angle (degrees), and starting height. Select a gravity preset (Earth, Moon, Mars, etc.) or enter custom gravity. Click Launch to calculate range, time of flight, maximum height, and final velocity. The trajectory is displayed on an interactive canvas with real-time physics simulation.
Yes! Toggle the Air Resistance checkbox to enable quadratic drag simulation. Adjust air density (ρ), drag coefficient (Cd), cross-sectional area (A), and mass (m). Choose from 15+ object presets: tennis ball, baseball, basketball, soccer ball, golf ball, arrow, frisbee, feather, balloon, and more. Each preset uses real-world physics parameters.
Absolutely! Select from 17 celestial bodies: Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Moon, Europa, Titan, Pluto, Ceres, Sun, and White Dwarf. Each has accurate surface gravity values. Try the Moon Golf scenario to see how a golf ball travels 400+ meters in lunar gravity!
Try these popular sports scenarios: Basketball Free Throw (7 m/s at 50°), Soccer Goal Kick (20 m/s at 25°), Golf Perfect Drive (70 m/s at 12°), Baseball Home Run (45 m/s at 30°). Each scenario includes realistic air resistance, object physics, and target placement.
Very accurate! Without air resistance, we use closed-form kinematics equations for exact results. With air resistance enabled, we use Euler numerical integration with small time steps (dt=0.005s) to solve the differential equations. All object presets use real-world values for mass, drag coefficient, and cross-sectional area from physics literature.

What is Projectile Motion?

θ
Max Height (H)
Range (R)
vx = v₀cosθ →
vy ↑

Projectile motion is the motion of an object thrown or projected into the air, subject only to gravity (and optionally air resistance). The object follows a curved path called a parabola. It is one of the most fundamental topics in classical mechanics, studied in physics courses from high school through university.

Velocity Decomposition

The initial velocity v₀ is split into two independent components: horizontal (vx = v₀cosθ) which stays constant, and vertical (vy = v₀sinθ) which changes due to gravity. This independence is the key insight of projectile motion.

The Three Key Equations

Range: R = v₀² sin(2θ) / g

Time of Flight: T = 2v₀ sin(θ) / g

Max Height: H = v₀² sin²(θ) / (2g)

The 45° Rule

Without air resistance and from ground level, a 45° launch angle maximizes range. Complementary angles (e.g., 30° and 60°) produce the same range but different trajectories. Air drag shifts the optimal angle below 45°.

Why does air resistance matter? In real life, the drag force Fd = ½ρCdAv² opposes motion and depends on the object’s speed, shape, size, and the air density. Light objects (like a feather) are dramatically slowed, while dense objects (like a cannonball) are barely affected. This calculator lets you toggle drag on/off to see the difference — try the Feather vs Cannonball guided experiment above!

On the Moon
Gravity is only 1.62 m/s² (vs Earth’s 9.81). A ball launched at 25 m/s travels ~6× farther on the Moon.
Energy Conservation
Without drag, KE + PE stays constant. At the apex, all kinetic energy has converted to potential energy, and vy = 0.
Real-World Applications
Sports science, military ballistics, space missions, civil engineering (bridge arcs), video game physics, and forensic analysis all use projectile motion.

About This Projectile Motion Calculator

Without Air Resistance: Uses closed-form kinematics equations (SUVAT formulas). Range: R = (v₀² × sin(2θ)) / g. Time of Flight: T = (2v₀ × sin(θ)) / g. Max Height: H = (v₀² × sin²(θ)) / (2g).

With Air Resistance: Euler numerical integration (dt=0.005s) solves quadratic drag: Fdrag = 0.5·ρ·Cd·A·v². Each object preset uses real-world values for mass, drag coefficient, and cross-sectional area.

Privacy: All calculations run 100% locally in your browser. No server uploads, no data collection, no registration required.

Author: Anish Nath | Last updated: 2025-11-19

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