Projectile Motion Calculator

Calculate range, maximum height, time of flight, and full trajectory for any projectile. Supports multiple planets, unit conversions, and an interactive SVG chart.

Presets:
degrees (0–90)
g = m/s²
degrees
m/s²

How Projectile Motion Works

Projectile motion describes the two-dimensional movement of an object launched into the air under the sole influence of gravity. The key insight is that horizontal and vertical motions are independent: horizontal velocity stays constant (no air resistance), while gravity continuously accelerates the object downward.

The resulting path is a parabola. This was first correctly described by Galileo Galilei in the early 17th century, overturning Aristotle's incorrect view that projectiles travel in straight lines before falling.

vx = const
Horizontal velocity never changes (no air resistance)
ay = −g
Vertical acceleration is constant downward at g m/s²
45°
Optimal launch angle for maximum range on flat ground

Projectile Motion Formulas

Velocity Components

vx = v₀ · cos(θ)

vy = v₀ · sin(θ)

Time of Flight (level ground)

T = 2 · vy / g

Double the time to peak

Range

R = vx · T

= v₀² · sin(2θ) / g

Maximum Height

H = vy² / (2g)

Height above launch point

Time of Flight (with height h₀)

T = (vy + √(vy²+2gh₀)) / g

Impact Velocity

v_f = √(vx² + vyf²)

vyf = vy − g·T

Range vs Angle Reference Table

For v₀ = 10 m/s on Earth (g = 9.81 m/s²)

Angle (θ) Range (m) Max Height (m) Flight Time (s)
15° 5.09 0.34 0.53
30° 8.83 1.27 1.02
45° (max range) 10.19 2.55 1.44
60° 8.83 3.83 1.77
75° 5.09 4.76 1.97

Real-World Applications

⚽

Sports

Footballs, basketballs, baseballs, javelins, and golf balls all follow projectile paths. Optimising launch angle maximises distance.

🚀

Military & Ballistics

Artillery shells, rockets, and missiles use projectile calculations adjusted for air resistance, spin, and wind.

🌊

Water Fountains

Engineers use projectile formulas to design the arc of water jets in fountains and irrigation systems.

🎢

Theme Parks

Water slide exit angles, roller coaster launches, and stunt show jumps are designed with projectile motion principles.

Worked Examples

Example 1 — Soccer kick (v₀ = 25 m/s, θ = 45°, Earth)

vx = 25 × cos(45°) = 17.68 m/s
vy = 25 × sin(45°) = 17.68 m/s
T = 2 × 17.68 / 9.81 = 3.60 s
R = 17.68 × 3.60 = 63.7 m
H = 17.68² / (2 × 9.81) = 15.9 m

Example 2 — Cannon on a cliff (v₀ = 100 m/s, θ = 30°, h₀ = 50 m)

vx = 100 × cos(30°) = 86.60 m/s
vy = 100 × sin(30°) = 50.00 m/s
T = (50 + √(2500 + 2×9.81×50)) / 9.81 = 11.01 s
R = 86.60 × 11.01 = 953 m
H = 50²/(2×9.81) + 50 = 177.6 m

Example 3 — Same throw on the Moon (v₀ = 25 m/s, θ = 45°, g = 1.62 m/s²)

T = 2 × 17.68 / 1.62 = 21.83 s
R = 17.68 × 21.83 = 385.9 m
H = 17.68² / (2 × 1.62) = 96.5 m
About 6× farther than on Earth!

Example 4 — Golf drive (v₀ = 70 m/s, θ = 12°, Earth)

vx = 70 × cos(12°) = 68.49 m/s
vy = 70 × sin(12°) = 14.56 m/s
T = 2 × 14.56 / 9.81 = 2.97 s
R = 68.49 × 2.97 = 203 m
H = 14.56² / (2 × 9.81) = 10.8 m

Frequently Asked Questions

Projectile motion is the curved path followed by an object launched into the air under the influence of gravity alone (ignoring air resistance). The horizontal velocity remains constant while the vertical velocity changes due to gravitational acceleration. Examples include thrown balls, launched rockets, and jumping athletes.
On a flat surface (launch and landing at the same height), a 45° launch angle gives the maximum range. This is because range R = v₀² × sin(2θ) / g, and sin(2θ) is maximised when 2θ = 90°, i.e., θ = 45°.
Air resistance (drag) reduces range and lowers the optimal launch angle below 45°. In real-world applications like golf or javelin, aerodynamic effects mean the actual optimal angle is typically 30–40°. This calculator uses the ideal model with no air resistance.
When launching from height h₀, time of flight T = (vy + √(vy² + 2·g·h₀)) / g, where vy = v₀·sin(θ) and g is gravitational acceleration. Use the Advanced tab in this calculator to include an initial height.
If the landing point is lower than the launch point, enter a positive initial height h₀ in the Advanced tab. The calculator will solve for the full time of flight using the quadratic formula, giving a longer range than a level launch.
Range (R) is the horizontal distance from launch to landing. Maximum height (H) is the highest vertical point above the launch level. Range depends on both speed and angle, while maximum height depends mainly on the vertical component of velocity: H = vy² / (2g).
Stronger gravity (larger g) shortens both range and time of flight proportionally. On the Moon (g ≈ 1.62 m/s²), the same throw travels about 6× farther than on Earth (g ≈ 9.81 m/s²). On Jupiter (g ≈ 24.79 m/s²), range shrinks to about 40% of Earth's value.
Because range R = v₀² × sin(2θ) / g, and sin(2×30°) = sin(60°) = sin(120°) = sin(2×60°). Complementary angles (summing to 90°) always produce identical ranges on level ground. However, 60° reaches a higher peak while 30° has a lower, flatter trajectory.

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