Projectile Motion Calculator

Projectile Motion Equations and How to Use Them

The projectile motion calculator analyzes two-dimensional motion under gravity (ignoring air resistance). At launch angle θ and initial velocity v₀, motion separates into independent horizontal (constant velocity) and vertical (constant acceleration g = 9.81 m/s²) components. "Projectile motion calculator" gets 1K monthly searches.

• Horizontal velocity (constant): vₓ = v₀ · cos(θ)
• Vertical velocity at time t: v_y(t) = v₀ · sin(θ) − g · t
• Horizontal position: x(t) = v₀ · cos(θ) · t
• Vertical position: y(t) = v₀ · sin(θ) · t − 1/2 · g · t²

Key Projectile Motion Formulas

• Time of flight: T = 2v₀ sin(θ) / g
• Maximum height: H = (v₀ sin θ)² / (2g)
• Horizontal range: R = v₀² sin(2θ) / g
• Maximum range (at θ = 45°): R_max = v₀² / g

Example: Ball launched at 20 m/s, angle 30°. Time of flight = 2 × 20 × sin(30°) / 9.81 = 2.04 s. Max height = (20 × 0.5)² / (2 × 9.81) = 5.10 m. Range = 20² × sin(60°) / 9.81 = 35.3 m.

Launch Angle and Range: Why 45° Maximizes Distance

• θ = 15°: Range = 0.5 × R_max (same as 75°)
• θ = 30°: Range = 0.866 × R_max (same as 60°)
• θ = 45°: Maximum range = v₀²/g. Equal horizontal and vertical components.
• θ = 60°: Same range as 30°, but higher arc and longer time of flight
• θ = 90°: No horizontal range; maximum height = v₀²/(2g)
• Complementary angles produce equal ranges: 25° and 65° land the same distance — different trajectories, same spot

Real-World Projectile Examples

• Basketball free throw: 4.5 m horizontal, launch at ~55°, ~7 m/s. Apex ~3.2 m above floor.
• Long jump: Takeoff angle ~20–22° (not 45° — horizontal speed more important than angle). World record ~8.95 m.
• Soccer penalty kick: Ball at ~15–25°, 25–30 m/s, reaches goal in ~0.5 seconds.
• Artillery: Long-range rounds fired at 45–60°. Flat-trajectory direct fire rounds at 5–15°.
• Javelin: Optimal angle ~35° due to air resistance (not 45°). World record ~90 m at ~30 m/s release.

Frequently Asked Questions About Projectile Motion

Does mass affect projectile motion?

In ideal projectile motion (no air resistance), mass has absolutely no effect. All objects accelerate downward at g = 9.81 m/s² regardless of mass — Galileo's famous result demonstrated by dropping objects from the Leaning Tower of Pisa. With air resistance (real-world), lighter/less dense objects decelerate more from drag relative to their weight, which is why a feather falls slower than a coin in air but both fall identically in a vacuum.

How does gravity on other planets change projectile range?

Range scales inversely with g: R = v₀² sin(2θ) / g. On Mars (g = 3.72 m/s²): range is 9.81/3.72 = 2.64× greater than on Earth with the same launch. On the Moon (g = 1.62 m/s²): range is 6.1× greater. Apollo 14 astronaut Alan Shepard famously hit a golf ball on the Moon in 1971 that traveled approximately 200 yards — impossible on Earth with a single-handed swing due to lunar gravity and the complete absence of air resistance.

What is the effect of initial height on projectile range?

When launched from an elevated position (cliff, building) above the landing point, the optimal angle for maximum range shifts below 45°. The additional height effectively gives the projectile more "fall time," so a shallower angle with more horizontal velocity becomes optimal. For a launch from height h, the optimum angle θ = arcsin(1/√(2 + 2gh/v₀²)) which is always less than 45° when h > 0. This is why a javelin thrower on a slope or an artillery piece firing downhill uses angles flatter than 45°.