Projectile Motion Calculator
Calculate range, max height, time of flight, and trajectory.
All calculations run live in your browser. Earth gravity = 9.80665 m/s² (NIST). Air resistance is not modeled.
What Is the Projectile Motion Calculator?
Projectile motion describes the two-dimensional path of an object launched with an initial velocity, subject only to gravity (air resistance is neglected). The horizontal and vertical motions are completely independent, horizontal velocity is constant; vertical velocity changes due to gravity.
- ›Horizontal motion, constant velocity. x(t) = vₓ·t where vₓ = v₀ cos θ. No force acts horizontally (in the ideal model), so the projectile covers equal horizontal distances in equal time intervals.
- ›Vertical motion, uniformly accelerated by gravity. y(t) = h₀ + v_y₀·t − ½g·t². The vertical speed decreases linearly until the apex, then increases downward.
- ›Initial height h₀, launching from a cliff, table, or hill adds initial height. The flight time increases, so range is greater than the level-ground case at the same angle.
- ›Gravity presets, choose Earth (9.80665 m/s²), Moon (1.62 m/s²), Mars (3.721 m/s²), or Jupiter (24.79 m/s²). On the Moon, a ball thrown at 45° travels over 6× farther than on Earth.
Formula
Kinematic Equations (no air resistance)
vₓ = v₀ cos θ
v_y₀ = v₀ sin θ
x(t) = vₓ · t
y(t) = h₀ + v_y₀ · t − ½ g t²
Key Derived Formulas
Time of flight T: solve h₀ + v_y₀T − ½gT² = 0
Max height H = h₀ + v_y₀² / (2g)
Range R = vₓ · T
Impact speed = √(vₓ² + v_yf²)
where v_yf = v_y₀ − g·T (vertical speed at impact)
| Symbol | Quantity | SI Unit |
|---|---|---|
| v₀ | Initial speed | m/s |
| θ | Launch angle | degrees (°) |
| h₀ | Initial height | m |
| g | Gravitational acceleration | m/s² |
| T | Time of flight | s |
| R | Horizontal range | m |
| H | Maximum height | m |
How to Use
- 1Enter the initial speed v₀ in m/s (must be positive).
- 2Enter the launch angle θ between 0° and 90°.
- 3Enter the initial height h₀ in metres (0 for ground-level launch).
- 4Select a gravity preset (Earth, Moon, Mars, Jupiter) or choose Custom and enter g in m/s².
- 5Press Calculate or hit Enter. An SVG trajectory chart appears along with 8 computed values.
- 6Press Clear to reset all fields and remove saved inputs.
Example Calculation
A football is kicked at 25 m/s at 30° from ground level on Earth:
Range-Maximizing Angle
On flat ground (h₀ = 0), maximum range is achieved at θ = 45°. With positive h₀, the optimal angle is less than 45°. The symmetric complement angles (e.g., 30° and 60°) give the same range on flat ground but differ in height and flight time.
Understanding Projectile Motion
Projectile Motion in the Real World
Projectile motion governs everything from thrown balls to artillery shells to planetary probes. While this calculator uses the ideal model (no air resistance), it accurately represents the underlying physics and is the correct model for homework problems, conceptual understanding, and approximate real-world estimates at moderate speeds and distances.
Gravity Comparison
| Body | g (m/s²) | g / g_Earth | Range at 45°, 50 m/s |
|---|---|---|---|
| Earth | 9.80665 | 1.00× | 255.0 m |
| Moon | 1.62 | 0.165× | 1543 m |
| Mars | 3.721 | 0.379× | 671 m |
| Jupiter | 24.79 | 2.53× | 100.7 m |
Frequently Asked Questions
Why is 45° the optimal launch angle for maximum range?
The 45° optimum comes directly from the range formula:
- ›Level ground formula, R = v₀² sin(2θ) / g. This is maximized when sin(2θ) = 1, which requires 2θ = 90° → θ = 45°.
- ›Complementary angles, sin(2×30°) = sin(60°) = sin(2×60°) = sin(120°). Angles 30° and 60° give identical range on flat ground. The 60° shot is higher and slower; the 30° shot is flatter and faster.
- ›With initial height h₀ > 0, the optimal angle drops below 45° because extra time in the air from height helps the lower-angle shot more. A cliff jump is better at a shallow angle.
- ›With air resistance, drag always reduces range and lowers the optimal angle. Competitive javelin throwers launch at ≈ 33–36°, not 45°, because of aerodynamic drag on the implement.
Why are horizontal and vertical motions treated independently?
The independence of horizontal and vertical motion is one of Galileo's key discoveries:
- ›No horizontal force, in the absence of air resistance, no force acts in the horizontal direction after launch. F=0 → a=0 → constant horizontal velocity.
- ›Vertical force only, gravity acts downward at g = 9.80665 m/s². This creates uniform downward acceleration regardless of horizontal speed.
- ›Galileo's experiment, a ball dropped straight down and a ball fired horizontally both reach the ground in exactly the same time (same vertical distance, same vertical acceleration).
- ›Mathematical consequence, the two differential equations are uncoupled: ẍ = 0 and ÿ = −g. They can be solved separately and combined to find the parabolic trajectory.
How does gravity differ on other planets?
Surface gravity varies widely across the solar system:
- ›Earth, g = 9.80665 m/s² (NIST standard). The reference for this calculator.
- ›Moon, g = 1.62 m/s² (1/6 of Earth). Alan Shepard drove a golf ball ≈ 200 m on the Moon in 1971. The same shot on Earth would travel ≈ 33 m.
- ›Mars, g = 3.721 m/s² (38% of Earth). Future Mars missions must account for different projectile ballistics in designing landing systems and sports.
- ›Jupiter, g = 24.79 m/s² (2.5× Earth). Range and flight time are dramatically reduced. A 45° shot at 50 m/s travels only 100 m on Jupiter vs. 255 m on Earth.
Why doesn't this calculator include air resistance?
The drag-free projectile model is pedagogically standard for good reasons:
- ›Exact solutions, without drag, range and height have clean closed-form formulas. Adding drag makes the equations non-linear ODEs with no simple analytic solution.
- ›Many free parameters, drag depends on drag coefficient C_d, cross-sectional area A, air density ρ, and mass m. These vary wildly: a bullet, a baseball, and a parachute all have very different drag characteristics.
- ›Curriculum standard, high school and university physics courses universally use the drag-free model for projectile motion problems. This calculator matches that standard.
- ›When drag matters, competitive ballistics, aeronautics, and sports science (baseball, golf, cricket) all require drag models. Those problems use specialized simulation software, not general-purpose calculators.
What is the relationship between launch angle and range?
The sin(2θ) factor in the range formula creates a symmetric, bell-shaped relationship:
- ›θ = 0° → Range = 0 (ball goes straight forward, hits ground immediately)
- ›θ = 15° → Range = v₀² sin(30°)/g = 0.5 × v₀²/g
- ›θ = 30° → Range = v₀² sin(60°)/g ≈ 0.866 × v₀²/g
- ›θ = 45° → Range = v₀² sin(90°)/g = v₀²/g (MAXIMUM)
- ›θ = 60° → Range = v₀² sin(120°)/g ≈ 0.866 × v₀²/g (same as 30°)
- ›θ = 75° → Range = v₀² sin(150°)/g = 0.5 × v₀²/g (same as 15°)
- ›θ = 90° → Range = 0 (ball goes straight up)
What happens when the initial height h₀ is greater than zero?
Launching from an elevated position significantly increases range:
- ›Longer flight time, the ball must fall through h₀ additional metres before landing. This extra descent takes extra time, during which the ball continues moving forward.
- ›Quadratic for T, the flight time equation h₀ + v_y₀T − ½gT² = 0 yields T = (v_y₀ + √(v_y₀² + 2g·h₀)) / g (taking the positive root).
- ›Optimal angle shifts, with h₀ > 0, the range-maximizing angle decreases below 45°. Intuitively, since the ball has "bonus time" from the height, it benefits more from a shallower, faster horizontal shot.
- ›Real examples, a cannon on a cliff, a soccer goalkeeper kick from ground, a baseball thrown from the pitcher's mound (slight elevation). In each case, h₀ slightly increases range compared to the level formula.
What is the trajectory chart showing?
The trajectory chart provides visual feedback about the parabolic path:
- ›Green dot, launch position at (0, h₀). The starting point of the trajectory.
- ›Blue dot, apex. The highest point of the trajectory, occurring at time t = v_y₀/g.
- ›Red dot, impact point at x = Range, y = 0 (ground level). The total horizontal distance traveled.
- ›Parabolic shape, the trajectory is a downward-opening parabola. Steeper angles create taller, shorter parabolae; shallower angles create flat, wide ones.
- ›60 sample points, the chart plots 60 evenly-spaced time samples from t=0 to t=T, then adds the final landing point for accuracy.