Projectile Motion Calculator — Range, Height & Flight Time

Calculate projectile range, maximum height, and time of flight instantly. Free physics calculator with initial velocity, launch angle, and step-by-step formula breakdown for parabolic trajectories.

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Projectile Motion Calculator

Enter launch parameters below to compute range, maximum height, time of flight, and velocity components.

Enter launch parameters and click Calculate Projectile Motion to see results.

Projectile Motion Formulas Explained

Projectile motion describes the curved path of an object launched near Earth's surface under constant gravitational acceleration, neglecting air resistance. The key kinematic formulas are:

Ground-Level Launch (h₀ = 0)

Range: R = v₀² × sin(2θ) / g
Max Height: H = v₀² × sin²(θ) / (2g)
Time of Flight: T = 2 × v₀ × sin(θ) / g

Elevated Launch (h₀ > 0)

Time of Flight: T = (v₀ sin(θ) + √(v₀² sin²(θ) + 2gh₀)) / g
Max Height: H = h₀ + v₀² sin²(θ) / (2g)
Range: R = v₀ cos(θ) × T

Variable Definitions

  • v₀ — Initial velocity (launch speed) in meters per second (m/s)
  • θ — Launch angle measured from horizontal in degrees
  • g — Gravitational acceleration (9.81 m/s² on Earth)
  • h₀ — Initial launch height above the landing surface
  • R — Horizontal range (distance traveled before landing)
  • H — Maximum height reached during flight
  • T — Total time of flight from launch to landing

How to Calculate Projectile Motion Step by Step

Follow these steps to manually compute all projectile motion parameters:

  1. Resolve initial velocity into components — v₀x = v₀ × cos(θ), v₀y = v₀ × sin(θ)
  2. Calculate time to reach maximum height — tmax = v₀y / g
  3. Determine maximum height — H = h₀ + v₀y² / (2g)
  4. Compute total time of flight — For ground-level: T = 2 × tmax. For elevated: use the quadratic formula.
  5. Calculate horizontal range — R = v₀x × T
  6. Verify with known values — At 45° on level ground, range is maximized at R = v₀² / g

Projectile Motion Worked Examples

Example 1: Ground-Level Launch at 45°

A ball is launched from ground level at 20 m/s at a 45° angle. Earth gravity g = 9.81 m/s².

v₀x = 20 × cos(45°) = 14.14 m/s
v₀y = 20 × sin(45°) = 14.14 m/s
tmax = 14.14 / 9.81 = 1.44 s
H = 14.14² / (2 × 9.81) = 10.19 m
T = 2 × 1.44 = 2.88 s
R = 14.14 × 2.88 = 40.77 m

Example 2: Elevated Launch from a 10 m Cliff

A projectile is launched from a 10 m cliff at 25 m/s at 30°.

v₀y = 25 × sin(30°) = 12.5 m/s
v₀x = 25 × cos(30°) = 21.65 m/s
H = 10 + 12.5² / (2 × 9.81) = 17.97 m
T = (12.5 + √(12.5² + 2×9.81×10)) / 9.81 = 3.11 s
R = 21.65 × 3.11 = 67.35 m

Example 3: Maximum Range Verification

At 45° on level ground, sin(2×45°) = sin(90°) = 1, confirming R = v₀²/g is the theoretical maximum range for a given launch speed.

Real-World Projectile Motion Applications

  • Sports Science: Analyzing basketball shots, golf drives, soccer kicks, and baseball trajectories to optimize launch angles.
  • Ballistics & Defense: Calculating artillery shell trajectories, missile paths, and bullet drop for military and forensic analysis.
  • Engineering: Designing water fountains, fireworks displays, and material ejection systems in manufacturing.
  • Space Exploration: Computing rocket launch profiles, lunar lander descents, and interplanetary trajectory planning.
  • Civil Engineering: Determining demolition blast trajectories and rockfall paths for highway safety barriers.
  • Education: Teaching Newtonian mechanics, kinematic equations, and vector decomposition in physics classrooms.
  • Game Development: Programming realistic physics engines for video game projectiles, grenades, and character jumps.

People Also Ask About Projectile Motion

Projectile motion is the two-dimensional motion of an object launched into the air, moving under constant gravitational acceleration while maintaining constant horizontal velocity (neglecting air resistance). The path follows a parabolic curve described by kinematic equations combining uniform horizontal motion with uniformly accelerated vertical motion.
For ground-level launches, range R = v₀² sin(2θ) / g. Maximum range occurs at 45°. For elevated launches, R = v₀ cos(θ) × T, where T is total flight time calculated using the quadratic time-of-flight formula.
On level ground with no air resistance, 45 degrees gives maximum range because sin(2θ) peaks at sin(90°) = 1. For elevated launches from a height, the optimal angle is slightly less than 45°, decreasing as launch height increases.
Maximum height H = h₀ + v₀² sin²(θ) / (2g). The term v₀² sin²(θ)/(2g) is the additional height gained above the launch point, derived from the vertical component of kinetic energy converting to gravitational potential energy.
Yes. Air resistance (drag) reduces range, maximum height, and flight time compared to ideal calculations. Real-world projectiles experience drag proportional to velocity or velocity squared depending on speed. This calculator uses idealized no-drag physics suitable for educational purposes.

Frequently Asked Questions

No. This calculator uses idealized parabolic motion without air resistance (drag). Results represent theoretical maximum values. For real-world applications involving significant air resistance, more complex drag models are required.
Yes. Toggle to "Elevated Launch" mode to input an initial height above the landing surface. The calculator then uses the general time-of-flight formula that accounts for the additional fall distance.
Use meters per second (m/s) for velocity, degrees for launch angle, meters (m) for height, and m/s² for gravity. The default gravity value is 9.81 m/s² (Earth). For the Moon, use 1.62 m/s²; for Mars, use 3.71 m/s².
Absolutely. Change the gravity input to 1.62 for lunar conditions, 3.71 for Mars, or any custom value. Lower gravity increases range, flight time, and maximum height proportionally.
At 90° (straight up), the range is zero because there is no horizontal velocity component. The projectile goes straight up and falls back to its launch point. Maximum height = h₀ + v₀²/(2g), and time of flight = 2v₀/g for ground-level launches.
The range formula R = v₀² sin(2θ)/g contains sin(2θ). The sine function reaches its maximum value of 1 when its argument is 90°, which occurs at θ = 45°. This balances horizontal distance coverage with time aloft.

Projectile Motion Glossary

Projectile Motion

The curved path of an object launched into the air under gravity, combining constant horizontal velocity with vertically accelerated motion.

Range

The total horizontal distance traveled by a projectile from launch point to landing point, calculated as R = v₀x × T.

Maximum Height

The highest vertical position reached during flight, occurring when vertical velocity momentarily becomes zero at the trajectory apex.

Time of Flight

The total duration from launch to landing. For ground-level symmetric trajectories, T = 2v₀ sin(θ)/g.

Launch Angle

The angle between the initial velocity vector and the horizontal plane, measured in degrees. Determines the trajectory shape.

Initial Velocity

The speed at which the projectile is launched, resolved into horizontal (v₀x) and vertical (v₀y) components.

Parabolic Trajectory

The characteristic U-shaped path of an ideal projectile, described by a quadratic equation in the absence of air resistance.

Gravitational Acceleration

The constant downward acceleration (g = 9.81 m/s² on Earth) acting on all projectiles near the planet's surface.

Editorial Review & Methodology

This projectile motion calculator was built and reviewed by the NumbrWiz Editorial Team. The formulas are derived from classical Newtonian mechanics and verified against standard physics textbooks including Halliday & Resnick's Fundamentals of Physics and the OpenStax University Physics curriculum.

  • Formula verification: All kinematic equations cross-checked against authoritative physics references and peer-reviewed educational resources.
  • Edge case testing: Validated at boundary angles (0°, 45°, 90°), zero initial height, elevated launches, and non-Earth gravity values.
  • UX review: Designed for intuitive physics input with clear unit labels, mode toggles, and comprehensive step-by-step breakdown.

Transparency note: All calculations run client-side in your browser. No data is ever collected, stored, or transmitted. This calculator uses idealized no-drag physics. For engineering or safety-critical applications, consult appropriate references and factor in air resistance, wind, and other real-world variables.

Page last reviewed: May 2026 · NumbrWiz Editorial Team