What is projectile motion?

Projectile motion is the motion of an object launched into the air that moves under the influence of gravity alone. The path it follows is a parabola, and the motion can be analysed by decomposing it into independent horizontal and vertical components.

Why does a 45-degree launch angle give the maximum range?

The range equation R = v^2 sin(2theta) / g is maximised when sin(2theta) = 1, which occurs at theta = 45 degrees. At this angle the horizontal and vertical velocity components are balanced for the greatest distance on flat ground.

How does air resistance affect a projectile's trajectory?

Air resistance (drag) reduces both the range and the maximum height of a projectile. It also breaks the symmetry of the parabolic path, causing the descent to be steeper than the ascent, and shifts the optimal launch angle below 45 degrees.

How do you calculate the maximum height of a projectile?

Maximum height H = v^2 sin^2(theta) / (2g), where v is initial speed, theta is launch angle and g is gravitational acceleration. The peak occurs when the vertical velocity component reaches zero.

What is the time of flight formula for projectile motion?

For a ground-level launch the time of flight is T = 2v sin(theta) / g. From a height h, T = (v sin(theta) + sqrt(v^2 sin^2(theta) + 2gh)) / g.

Projectile Motion Simulator

Trajectory • Range • Max Height • Time of Flight — Simulate • Explore • Practice • Quiz

Mode

Units

📖 User Guide

💡 Tip: click and drag from the launch point on the canvas to aim & set speed.

Angle

Velocity

m/s

Height

Gravity

Presets

Show Grid Velocity vectors Range/height markers Ghost trails

Air Resistance

Range

0.00 m

Max Height

0.00 m

Time of Flight

0.00 s

Impact Velocity

0.00 m/s

Launch Angle

45 °

Initial Velocity

30 m/s

📖 Learning panels

Σ Live equations — values substituted from current state

⇢ Velocity components — horizontal & vertical split

💡 What-if coach — insights from current values

User Guide — Projectile Motion Simulator

1 Overview

This free projectile motion calculator online lets you launch virtual projectiles at any angle and velocity, then watch the parabolic path unfold in real time on an interactive canvas. The simulator computes range, maximum height, time of flight, and impact velocity using standard kinematic equations. It supports three gravity environments (Earth, Moon, Mars), optional air resistance, elevated launch heights, an SI/Imperial unit toggle, undo/redo, and CSV/PNG export — making it a complete trajectory analysis tool.

Whether you are learning about velocity components, complementary launch angles, or the effect of drag on a projectile’s trajectory, this tool provides instant visual feedback backed by precise numerical readouts. No downloads or accounts required.

2 Getting Started

The simulator loads in Simulate mode with a default launch angle of 45° and velocity of 30 m/s on Earth (g = 9.81 m/s²). The canvas shows the launcher, ground plane, and trajectory grid. Six readout cards display Range, Max Height, Time of Flight, Impact Velocity, Launch Angle, and Initial Velocity.

Switch between the four modes using the pill tabs: Simulate for hands-on launching, Explore for concept study, Practice for random calculation problems, and Quiz for timed multiple-choice questions. Use the SI / Imperial toggle in the top bar to switch every reading (distance, velocity, height) between metric and US customary units.

3 Simulate Mode

Use either the slider or the [−] [input] [+] stepper next to each parameter to set Angle (0–90°), Velocity (5–100 m/s, auto-converted when Imperial is selected), and Height (0–50 m). Choose Earth, Moon, or Mars gravity. Toggle Air Resistance on to see how drag shortens range and breaks trajectory symmetry.

Press Fire (or Space) to launch. The animation traces the trajectory in real time, and all readout cards update at impact. The Show Calculations button on the canvas opens a step-by-step modal proving the result in SI. Previous trajectories remain as ghost trails for side-by-side comparison.

The Show row of checkboxes lets you hide Grid, Velocity vectors, Range/height markers, and Ghost trails to focus on a specific concept. Use the presets — Basketball Shot, Artillery, Golf Drive, Optimal 45° — to explore realistic scenarios.

4 Learning Panels, Export & Right-Click

Below the Simulate controls, the Learning panels section provides three collapsible cards: Live equations shows the range, height and time-of-flight formulas with your current values substituted in classical KaTeX notation; Velocity components breaks v into V_x and V_y0; and What-if coach generates plain-English insights about your shot.

Click CSV to download the full trajectory (x, y, V_x, V_y, t) as a spreadsheet, or PNG to save the canvas image with a watermark. Right-click anywhere on the canvas for a context menu with Export, Toggle Grid, and Reset options.

Keyboard shortcuts: Space or Enter = Fire, R = Reset, ↑↓ = adjust angle ±1°, ←→ = adjust velocity ±1, Ctrl+Z = Undo, Ctrl+Shift+Z = Redo.

5 Explore Mode

Explore mode contains 12 concept cards sorted into three categories: Kinematics, Vectors, and Applications. Each card covers one topic — such as the range equation, velocity components, maximum height formula, or the effect of air resistance — with a definition, formula, and a worked numerical example.

This mode is ideal for building a solid conceptual foundation before attempting calculations. Use the category pills to filter topics, then click any card to open its detailed explanation.

6 Practice & Quiz

Practice mode generates unlimited random problems covering range, maximum height, time of flight, and velocity components. Enter your answer and press Check. If incorrect, the full step-by-step solution appears so you can identify your mistake. Your running score tracks accuracy across the session.

Quiz mode presents 5 randomised questions per session mixing conceptual and numerical problems. Topics span everything from trajectory symmetry to the independence of horizontal and vertical motion. A detailed score breakdown is displayed at the end of each quiz.

7 Tips & Best Practices

Fire at complementary angles (e.g., 30° then 60°) to verify they produce the same range but different heights and flight times.
Compare with and without air resistance to understand how drag reduces range and shifts the optimal angle.
Try Moon gravity to see how dramatically lower g increases range — the same launch on the Moon travels roughly six times farther.
Switch to Imperial when working with US textbook problems (ft, ft/s).
Open Show Calculations to step through the derivation in SI before checking your hand-worked answer.
The simulator works fully offline once loaded — ideal for classroom use without reliable internet.

Understanding Projectile Motion — Free Interactive Simulator

Projectile motion describes an object moving through the air under gravity alone, tracing a parabolic path. This free simulator lets you launch projectiles at any angle and speed, then computes range, maximum height, and time of flight from the standard kinematic equations — with optional air resistance and lunar or Martian gravity.

What is the range formula for projectile motion?

For a projectile launched from ground level with initial velocity v at angle θ, the horizontal range is R = v²·sin(2θ) / g. The maximum range on flat ground occurs at 45°, because sin(2×45°) = 1. Complementary angles such as 30° and 60° produce the same range but with different heights — the lower angle gives a flatter, faster path and the higher angle a taller, slower arc.

Key Projectile Motion Equations (Featured Snippet)

Quantity	Formula (SI)	Symbol & Unit
Range (ground launch)	R = v²·sin(2θ) / g	R, m
Maximum height	H = v²·sin²(θ) / (2g)	H, m
Time of flight (ground)	T = 2v·sin(θ) / g	T, s
Time to peak	t_p = v·sin(θ) / g	t_p, s
Horizontal velocity	V_x = v·cos(θ)	V_x, m/s
Vertical velocity	V_y = v·sin(θ) − g·t	V_y, m/s
Impact speed (from height)	v_f = √(v² + 2gh)	v_f, m/s
Drag force (with air)	F_d = ½·C_d·ρ·A·v²	F_d, N

How do you calculate maximum height and time of flight?

The maximum height reached by a projectile is H = v²·sin²(θ) / (2g). At this point, the vertical velocity component becomes zero while the horizontal component continues unchanged. The time of flight for a ground-level launch is T = 2v·sin(θ) / g. When launching from height h, the time of flight increases because the projectile has farther to fall, and the range increases accordingly.

Why are horizontal and vertical motion independent?

The horizontal velocity V_x = v·cos(θ) remains constant throughout the flight (in the absence of air resistance), while the vertical velocity V_y = v·sin(θ) − gt changes linearly due to gravitational acceleration. Because gravity acts only vertically, the two motions evolve independently — a bullet fired horizontally and one dropped from the same height strike the ground at exactly the same time. The impact velocity follows from the Pythagorean theorem on the final velocity components.

How does air resistance change the trajectory?

In reality, air resistance (drag) significantly affects projectile trajectories. The drag force F_d = ½·C_d·ρ·A·v² acts opposite to the velocity vector, reducing both range and maximum height while breaking the symmetry of the parabolic path. The optimal launch angle shifts below 45° when drag is present. This simulator lets you toggle air resistance on and off to compare ideal and realistic trajectories side by side.

How do I use this projectile motion simulator?

In Simulate mode, adjust angle, velocity, and height using sliders or steppers, then press Fire to watch the projectile trace its trajectory. Switch between Earth, Moon, and Mars gravity, toggle Air Resistance, or flip to Imperial units. Open Show Calculations for a step-by-step derivation in SI. Use Explore mode for 12 concept cards, Practice for unlimited random problems, and Quiz for 5-question timed assessments.

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