Projectile Motion Calculator - Free Online Physics Tool

Projectile Motion Calculator

Calculate range, height, time of flight, and trajectory for any projectile

Initial Velocity (v₀)

m/s

km/h

mph

Launch Angle (θ)

Common angles:

30°

45°

60°

75°

Initial Height (h₀)

Gravity (g)

Projectile Mass

Maximum Range

40.77 m

Optimal angle for maximum range

Time of Flight

2.88 s

Maximum Height

10.19 m

Impact Velocity

20.00 m/s

Projectile Trajectory

Trajectory visualization would appear here

Parabolic path: y = h₀ + x·tanθ - (g·x²)/(2·(v₀·cosθ)²)

Projectile Motion Formulas

Range: R = (v₀²·sin2θ)/g

Time of flight: t = (2v₀·sinθ)/g

Max height: H = h₀ + (v₀²·sin²θ)/(2g)

Horizontal velocity: vₓ = v₀·cosθ (constant)

Vertical velocity: vᵧ = v₀·sinθ - g·t

Trajectory equation: y = h₀ + x·tanθ - (g·x²)/(2·(v₀·cosθ)²)

People Also Ask

🤔 What is projectile motion and how is it calculated?

Projectile motion is the motion of an object thrown or projected into air, subject only to gravity. It's calculated using kinematic equations with constant acceleration g downward.

🔍 What angle gives maximum range for a projectile?

45° gives maximum range when launched from ground level. With initial height, optimal angle is less than 45°. Range is maximum when sin2θ is maximum (θ=45°).

⚡ How does air resistance affect projectile motion?

Air resistance reduces range, height, and time of flight. Trajectory is no longer perfectly parabolic. For accurate results in real world, aerodynamic drag must be considered.

📏 What's the difference between range and maximum height?

Range is horizontal distance traveled. Maximum height is highest vertical point reached. 45° maximizes range, 90° maximizes height (straight up), but gives zero range.

🎯 How do I calculate projectile motion with initial height?

Add initial height h₀ to height equations. Time of flight becomes: t = [v₀·sinθ + √((v₀·sinθ)² + 2gh₀)]/g. Use our calculator with height > 0.

🔥 What are real-world applications of projectile motion?

Sports (basketball, golf), military (artillery), fireworks, space launches, water fountains, and any object thrown, kicked, or launched through air.

What is Projectile Motion?

Projectile motion is the motion of an object thrown or projected into the air, subject only to the acceleration of gravity (assuming no air resistance). The object is called a projectile, and its path is called its trajectory. Projectile motion is a form of two-dimensional motion or motion in a plane.

Key Characteristics of Projectile Motion:

1. Parabolic trajectory (when air resistance is negligible)
2. Constant horizontal velocity (no horizontal acceleration)
3. Constant vertical acceleration (g = 9.81 m/s² downward)
4. Independent horizontal and vertical motions (can be analyzed separately)

Assumptions in ideal projectile motion:

No air resistance - projectile moves in vacuum
Constant gravity - g doesn't change with height
Flat Earth - curvature and rotation effects ignored
Uniform gravitational field - g is constant in magnitude and direction
Point mass projectile - no rotation or size effects

How to Use This Calculator

Enter initial conditions to calculate complete projectile motion parameters:

Four Main Inputs:

Initial Velocity (v₀): Speed at launch point (m/s, km/h, or mph)
Launch Angle (θ): Angle from horizontal (0° to 90°)
Initial Height (h₀): Height above landing surface (0 for ground launch)
Gravity (g): Acceleration due to gravity (Earth, Moon, Mars, or custom)

The calculator instantly provides:

Range: Total horizontal distance traveled
Time of Flight: Total time in air
Maximum Height: Highest point above launch level
Impact Velocity: Speed when hitting ground
Trajectory Equation: Mathematical description of path
Unit conversions between metric and imperial systems
Gravity settings for different celestial bodies

Common Projectile Motion Examples

Here are typical projectile motion scenarios with calculated values (g = 9.81 m/s²):

Scenario	Velocity	Angle	Range	Max Height	Time
Basketball shot	8 m/s	60°	5.66 m	2.45 m	1.41 s
Football kick	25 m/s	45°	63.70 m	15.92 m	3.61 s
Golf drive	70 m/s	12°	170.8 m	8.99 m	2.48 s
Water fountain	15 m/s	80°	7.77 m	11.32 m	3.01 s
Catapult shot	40 m/s	30°	141.4 m	20.39 m	4.08 s
Moon golf shot	70 m/s	45°	1,730 m	433 m	61.3 s

Optimal Angle Fact:

For maximum range from ground level: θ = 45°. For maximum range with initial height h₀: θ < 45°. Complementary angles (30° and 60°, 15° and 75°) give same range (from ground level) but different heights and times.

Common Questions & Solutions

Below are answers to frequently asked questions about projectile motion calculations and applications:

Calculation & Physics

Why is 45° optimal for maximum range? What's the proof?

The range equation is R = (v₀²·sin(2θ))/g. Since sin(2θ) has maximum value of 1 when 2θ = 90°, therefore θ = 45° gives maximum range.

Mathematical Proof:

Range R(θ) = (v₀²/g)·sin(2θ)

dR/dθ = (2v₀²/g)·cos(2θ) = 0 for maximum

cos(2θ) = 0 → 2θ = 90° → θ = 45°

This assumes launch from ground level. With initial height, optimal angle is less than 45°.

Practical note: In sports, optimal angle is often less than 45° due to air resistance, height of release, and target height. Baseball optimal ~30°, basketball ~48°, golf ~12°.

How do I calculate projectile motion with air resistance?

Air resistance (drag) makes calculations complex, requiring numerical methods:

Drag Force Equations:

F_drag = ½·ρ·v²·C_d·A

Where: ρ = air density, C_d = drag coefficient, A = cross-sectional area

Horizontal: dvₓ/dt = -(ρ·C_d·A/(2m))·v·vₓ

Vertical: dvᵧ/dt = -g - (ρ·C_d·A/(2m))·v·vᵧ

These differential equations require numerical integration (Euler, Runge-Kutta methods).

Effects of air resistance: Reduces range (20-50% for sports balls), lowers trajectory, makes landing angle steeper. Terminal velocity limits maximum speed for falling objects.

Practical Applications

How do athletes use projectile motion physics?

Understanding projectile motion gives athletes competitive advantages:

Sport	Optimal Angle	Key Physics	Strategy
Basketball	45-48°	Release height matters (2m)	Higher arc = larger target area
Football (soccer)	20-30°	Air resistance significant	Knuckleballs, curves via spin
Golf	10-15°	Maximize carry distance	Low launch + backspin = longer roll
Baseball	25-35°	Home run optimization	Launch angle + exit velocity critical
Shot Put	38-42°	Release height ~2.1m	Less than 45° due to height
Long Jump	20-25°	Take-off velocity ~9-10 m/s	Horizontal speed prioritized

Modern sports use high-speed cameras and computer analysis to optimize launch parameters. Baseball's "launch angle revolution" increased home runs by optimizing to 25-30°.

How is projectile motion used in military and space applications?

Projectile motion principles are critical for artillery, missiles, and space launches:

Military & Space Applications:

Artillery: Range tables account for air density, wind, Earth rotation (Coriolis effect)
Missile guidance: Real-time trajectory corrections using GPS and inertial navigation
Space launches: Rocket equations combine with projectile motion for orbital insertion
Ballistic missiles: Suborbital trajectories with ranges up to 15,000 km
Satellite deployment: Precise velocity and angle calculations for specific orbits
Re-entry vehicles: Ballistic coefficients for atmospheric re-entry

Example: ICBM with 7 km/s velocity, 25° launch angle has range ≈ 10,000 km. Atmospheric drag reduces this to practical ranges of 5,000-15,000 km.

Advanced Concepts

What happens at very high velocities (near orbital velocity)?

At velocities approaching orbital velocity (~7.8 km/s for Earth), projectile motion merges with orbital mechanics:

Orbital Velocity Thresholds:

Suborbital: v < 7.8 km/s - returns to Earth (ballistic missiles)
Low Earth Orbit: v ≈ 7.8 km/s - circular orbit just above atmosphere
Escape velocity: v ≥ 11.2 km/s - leaves Earth's gravity well
Geostationary: v ≈ 3.07 km/s at 35,786 km altitude

Projectile equations fail at orbital speeds because they assume flat Earth and constant g. Need orbital mechanics (Kepler's laws).

Interesting fact: If you throw a ball at 7.9 km/s horizontally from sea level (ignoring air), it would orbit Earth at ground level! Actually air friction would vaporize it instantly.

How does projectile motion differ on other planets/moons?

Different gravity significantly affects projectile motion:

Celestial Body	Gravity (m/s²)	Range Multiplier	Time Multiplier	Human Jump Example
Earth	9.81	1×	1×	0.5 m vertical jump
Moon	1.62	6.06×	2.46×	3.0 m vertical jump
Mars	3.71	2.64×	1.62×	1.3 m vertical jump
Jupiter	24.79	0.40×	0.63×	0.2 m vertical jump
Pluto	0.62	15.8×	3.98×	7.9 m vertical jump

Range is inversely proportional to g. Moon's low gravity (1/6 Earth) means 6× longer jumps. On Jupiter (2.5× Earth gravity), you could barely jump. Use our calculator's gravity selector to experiment!