Projectile Motion Calculator
Projectile motion describes the curved path of an object launched at an angle under gravity (ignoring air resistance). Enter the launch speed and angle to compute horizontal range, maximum height, and total flight time.
Advertisement
Calculator
See your Projectile Motion Calculator results
Enter your email to unlock results — free forever.
No spam, ever. Unsubscribe at any time.
Advertisement
Formula
R = v₀² sin(2θ) / g
Horizontal range R = v₀² × sin(2θ) / g. Maximum height H = v₀² × sin²(θ) / (2g). Flight time T = 2v₀ × sin(θ) / g. These formulas assume flat ground, no air resistance, and launch at ground level. θ is the launch angle; v₀ is initial speed; g = 9.81 m/s². Note: sin(2×45°) = 1, so 45° gives maximum range.
How to use the Projectile Motion Calculator
- 1
Enter your initial speed
Value should be in m/s.
- 2
Enter your launch angle
Value should be in °.
- 3
Enter your gravity
Value should be in m/s².
- 4
Read your results instantly
Results update in real time as you type.
Advertisement
Why 45° gives maximum range
The range formula R = v₀² sin(2θ) / g is maximized when sin(2θ) = 1, which occurs at 2θ = 90°, or θ = 45°. At this angle, horizontal and vertical velocity components are equal, creating the optimal balance between distance traveled horizontally and time in the air.
Interestingly, launch angles that add to 90° give the same range: 30° and 60°, 20° and 70°, etc. They cover the same horizontal distance but via different trajectories — the steeper angle reaches greater height and takes longer. This symmetry comes from the sin(2θ) term: sin(2×30°) = sin(60°) = sin(120°) = sin(2×60°).
Real projectiles vs. ideal
This calculator assumes no air resistance and flat ground. Real projectiles deviate significantly. A baseball thrown at 40 m/s would travel about 163 m in a vacuum at 45° — real batted balls rarely exceed 120 m due to drag. Artillery shells at high velocities experience severe drag and must account for the curvature and rotation of the Earth for long ranges.
Air resistance changes the optimal angle too: for a golf ball, the optimal angle is closer to 25-30° because drag is significant. The dimples on a golf ball reduce drag, increasing range by up to 50% compared to a smooth ball.
Advertisement
Applications of projectile motion
Projectile motion analysis is used in ballistics, sports science, aerospace engineering, and video game physics. In basketball, the optimal launch angle for a free throw depends on the height of release and distance to the basket — typically around 45-55°. In soccer, corner kicks and long passes require accurate projection calculations.
In engineering, the principles apply to water fountains (nozzle angles for maximum distance), sprinkler systems, and pneumatic conveying of granular materials. Space missions use related equations for orbital mechanics, where projectile motion merges with orbital dynamics at high velocities.
Tips & Insights
45° for maximum range on flat ground
On level ground, 45° always gives the maximum horizontal range for a given initial speed (ignoring air resistance).
Complementary angles give equal range
30° and 60°, or 20° and 70°, give the same range. The higher angle reaches greater altitude and stays airborne longer.
This formula assumes launch at ground level
If launching from a height (like throwing off a cliff), the formulas are more complex and this calculator won't give accurate results.
Worked Examples
Soccer free kick
Range ≈ 25² × sin(60°) / 9.81 ≈ 55.2 m. Max height ≈ 7.97 m. Flight time ≈ 2.55 s.
Maximum range throw
Range = 15² / 9.81 ≈ 22.9 m. Max height ≈ 5.74 m. Flight time ≈ 2.16 s.
Advertisement
Frequently Asked Questions
What launch angle gives maximum range?
45° gives maximum range on flat ground with no air resistance. With air resistance, the optimal angle is typically lower (25-35° for most projectiles).
Does mass affect projectile range?
In this ideal model, no — mass cancels out. In reality, heavier objects are less affected by air resistance relative to their momentum, so they often travel farther.
What is the shape of a projectile's path?
A parabola (in the absence of air resistance). Horizontal velocity is constant; vertical velocity changes due to gravity.
Why do complementary angles give the same range?
Because sin(2θ) = sin(180° - 2θ). For θ = 30° and 60°: sin(60°) = sin(120°). Same range, different trajectories.
How does this apply to basketball?
A basketball shot follows projectile motion from release point to hoop. The optimal arc depends on release height and distance. Higher arcs (50-55°) give larger entry windows into the hoop.
Advertisement