Range, Height & Flight Time
Last reviewed: January 2026
The Projectile Motion Calculator is a free browser-based tool that performs this calculation instantly with no signup or downloads required. Enter your values, click calculate, and get accurate results immediately. All processing happens in your browser — nothing is sent to a server.
Horizontal and vertical motion are independent. Horizontal velocity is constant (no air resistance). Vertical velocity changes due to gravity. Maximum range on level ground occurs at 45°. Maximum height occurs at 90° (straight up). Key equations: Range = v₀²sin(2θ)/g. Max Height = v₀²sin²(θ)/(2g). Time of flight = 2v₀sin(θ)/g. Real-world projectiles are affected by air resistance, spin (Magnus effect), and wind — this calculator assumes ideal conditions.
| Variable | Formula | Units |
|---|---|---|
| Range | R = v₀²sin(2θ)/g | meters |
| Max height | H = v₀²sin²(θ)/(2g) | meters |
| Time of flight | T = 2v₀sin(θ)/g | seconds |
| Optimal angle | 45° (max range) | degrees |
Projectile motion describes the path of an object launched into the air and subject only to gravity and its initial velocity (ignoring air resistance in the idealized case). The key insight, first described by Galileo, is that horizontal and vertical motions are independent — an object projected horizontally from a table hits the floor at the same time as an object dropped from the same height. This independence allows projectile motion to be analyzed as two separate one-dimensional problems: constant-velocity horizontal motion and constant-acceleration vertical motion under gravity (g ≈ 9.81 m/s² or 32.2 ft/s²).
The resulting trajectory is always a parabola in the absence of air resistance. The equations governing projectile motion are: horizontal position x = v₀ cos(θ) × t, vertical position y = v₀ sin(θ) × t - (1/2)g t², horizontal velocity vₓ = v₀ cos(θ) (constant), and vertical velocity vᵧ = v₀ sin(θ) - g × t. From these equations, key quantities can be derived: maximum height H = (v₀ sin θ)² / (2g), time of flight T = 2v₀ sin(θ) / g, and horizontal range R = v₀² sin(2θ) / g. The range formula reveals that maximum range occurs at a launch angle of 45° (since sin 90° = 1 is the maximum of sin 2θ), a result that has practical implications for sports, artillery, and engineering.
The relationship between launch angle and trajectory characteristics reveals elegant symmetries. Complementary angles (pairs that sum to 90°) produce the same horizontal range — a projectile launched at 30° travels the same horizontal distance as one launched at 60° with the same initial speed, though their trajectories look very different. The 30° launch produces a low, fast trajectory with shorter time of flight, while the 60° launch creates a high, arching path with longer time of flight and greater maximum height. At 45°, the range is maximized because the horizontal and vertical velocity components are equal, optimizing the tradeoff between how far the projectile travels horizontally and how long it stays in the air.
In practice, the optimal launch angle differs from 45° due to factors the idealized model ignores. Air resistance reduces the optimal angle to approximately 35-42° for most real projectiles because drag disproportionately affects high trajectories (which spend more time in the air). Launch height above the landing surface also changes the optimal angle — launching from an elevated position (a cliff, a basketball player's hands above head height) lowers the optimal angle below 45°. In sports, the optimal throwing angle for shot put is approximately 37-42° because the shot is released from above shoulder height, while a long jump takeoff angle is 18-23° because higher angles would reduce the horizontal velocity component too much. Our Gravitational Force Calculator explores the force driving projectile motion.
Real projectiles experience air resistance (drag) that significantly alters their trajectories from the idealized parabolic path. Drag force is proportional to velocity squared (F_drag = ½ρCdAv²), meaning it increases dramatically with speed and disproportionately affects the ascending portion of the trajectory (when velocity is highest). This causes real trajectories to be asymmetric — the descending path is steeper than the ascending path, and the projectile lands at a steeper angle than it was launched. The range of a real projectile is always less than the ideal calculation predicts, with the reduction most significant for lightweight, large-surface-area, high-speed objects.
The effect of air resistance varies enormously between different projectiles. A cannonball, being dense and relatively small, follows a nearly parabolic path. A baseball experiences significant drag and the Magnus effect (spin-induced curved trajectories), which pitchers exploit for curveballs and sliders. A shuttlecock in badminton has such high drag relative to its mass that its trajectory barely resembles a parabola. Golf balls use dimpled surfaces to create a turbulent boundary layer that actually reduces drag compared to a smooth ball, and backspin generates lift through the Magnus effect, allowing drives to travel 200-300 yards — far more than a smooth ball could achieve.
Projectile motion principles apply across diverse fields. In sports science, optimizing throwing, kicking, and hitting techniques requires understanding how launch angle, velocity, and spin interact. In forensic ballistics, trajectory reconstruction helps determine shooting positions and bullet paths. In military applications, artillery firing tables use sophisticated projectile motion calculations that account for drag, wind, Earth's rotation (Coriolis effect), temperature, altitude, and atmospheric pressure. In civil engineering, projectile motion governs the design of water fountains, irrigation sprinklers, and fire hose trajectories.
Space exploration relies on projectile motion at its most extreme — orbital mechanics is essentially projectile motion where gravity curves the trajectory into an ellipse rather than a parabola (because the Earth's surface curves away beneath the projectile). An object launched horizontally at approximately 7.9 km/s from the Earth's surface would, in the absence of air resistance, follow a trajectory where it perpetually falls toward Earth but never reaches the ground because the surface curves away at the same rate — this is orbital velocity, and it is the basis of all satellite and space station operations. For related physics calculations, see our Speed of Sound Calculator and Torque Calculator.
For long-range projectiles — such as artillery shells, long-distance rockets, and intercontinental ballistic missiles — the Coriolis effect becomes a significant factor. Because the Earth rotates, a projectile in flight maintains its initial angular momentum while the ground beneath it rotates, causing an apparent deflection to the right in the Northern Hemisphere and to the left in the Southern Hemisphere. For a typical artillery shell fired at medium range, the Coriolis deflection amounts to a few meters, which is tactically significant. For ICBMs traveling thousands of kilometers, the deflection can reach several kilometers. Snipers firing at extremely long ranges (1,000+ meters) must also account for the Coriolis effect, as even the small deflection over their bullet's flight time can cause a miss at such distances.
The basic projectile motion equations assume no air resistance and uniform gravity — useful approximations but not perfect. Air resistance (drag) significantly affects fast-moving objects: a baseball thrown at 90 mph travels roughly 30% less distance than the ideal calculation predicts. The drag force increases with the square of velocity, so it matters more for faster projectiles. The Magnus effect causes spinning objects to curve — topspin on a tennis ball makes it dip faster, while backspin extends hang time. At very long ranges (artillery, rockets), the Coriolis effect from Earth's rotation deflects trajectories. For everyday physics problems, the idealized model works well for speeds under ~20 m/s and ranges under ~100 meters. Explore the underlying force calculations with our Newton's Law Calculator.
→ Check your units. Science calculations are unit-sensitive. Make sure all inputs use the same system (SI or Imperial) to avoid conversion errors.
→ Understand the assumptions. Most science calculators use idealized models. Real-world conditions (friction, air resistance, impurities) may cause your results to differ from calculated values.
→ Use for estimation and verification. These tools are excellent for quick estimates and checking homework — but always show your work for academic submissions.
→ Explore the relationships. Change one variable at a time to see how it affects the result. This builds physical intuition about the underlying principles.
See also: Kinetic Energy Calculator · Speed Distance Time Calculator · Gravitational Force Calculator