Projectile Motion Calculator

The four essential equations (assuming launch from ground level, no air resistance):

• Range R = v₀²sin(2θ)/g — maximum at θ = 45°.
• Maximum height H = v₀²sin²(θ)/(2g).
• Time of flight T = 2v₀sin(θ)/g.

These decompose the launch velocity into components: v_x = v₀cos(θ) remains constant throughout flight; v_y = v₀sin(θ) - gt decreases until zero at peak, then increases downward.

At any time t: horizontal position x = v₀cos(θ)t, vertical position y = v₀sin(θ)t - ½gt².

Example: a cannonball fired at 100 m/s at 45°: R = 100²×sin(90°)/9.81 = 1,019 m, H = 100²×sin²(45°)/(2×9.81) = 255 m, T = 2×100×sin(45°)/9.81 = 14.4 seconds.

In a vacuum, maximum range occurs at 45° for any launch speed. But complementary angles (e.g., 30° and 60°) produce the same range — 30° gives a flat, fast trajectory while 60° gives a high, slow one.

For a launch from elevation (cliff, hill), the optimal angle shifts below 45° because the projectile has extra vertical distance to travel. From a height h, optimal angle ≈ arctan(v₀/√(v₀² + 2gh)), which approaches 45° as v₀ increases relative to h.

With air resistance, optimal angle drops to 30-40° for most real projectiles because high trajectories spend more time at high altitude where air resistance decumulates.

Artillery calculations must also account for:

• wind,
• the Coriolis effect (Earth's rotation),
• and aerodynamic properties of the shell.

Sports applications: a basketball free throw (initial speed ~7 m/s, angle ~52°, release height ~2.4 m, hoop at 3.05 m, distance 4.6 m) has a narrow window of successful angles and speeds. A golf drive (initial speed 70 m/s, angle 10-12° with backspin) uses the Magnus effect (spin-induced lift) to achieve ranges of 250+ meters — far exceeding the vacuum prediction at that angle.

Air resistance (drag force Fd = ½ρv²CdA) significantly affects trajectory: a baseball hit at 45 m/s at 35° travels approximately 120 meters in a vacuum but only 95-100 meters with air resistance — a 20% reduction. Drag force increases with the square of velocity, so faster projectiles are proportionally more affected.

Terminal velocity occurs when drag equals gravity:

• approximately 30 m/s for a baseball,
• 55 m/s for a skydiver (spread position),
• and 90 m/s for a skydiver (head-down).

To calculate horizontal range on flat ground, use R = v₀²·sin(2θ)/g, where v₀ is initial speed, θ is launch angle, and g is gravitational acceleration (9.81 m/s² on Earth).

• First square the initial velocity,
• then multiply by the sine of twice the launch angle,
• and divide by g.

For example, a ball launched at 25 m/s at 30°: R = 25² × sin(60°) / 9.81 = 625 × 0.8660 / 9.81 ≈ 55.2 meters. The doubled-angle term explains why 30° and 60° give equal range.

HyperPhysics (Georgia State University) derives this result from the independent horizontal and vertical motion equations.

Projectile motion uses SI base and derived units defined by the BIPM and NIST.

• Velocity (v₀) is measured in meters per second (m/s),
• launch angle (θ) in degrees or radians,
• distance and height (R, H) in meters (m),
• and time of flight (T) in seconds (s).

Gravitational acceleration g has units of meters per second squared (m/s²), with the standard value 9.80665 m/s² per NIST.

Keeping every quantity in consistent SI units guarantees the formulas return meters and seconds directly. If you input velocity in km/h or feet, convert first: 1 km/h = 0.2778 m/s, and 1 foot = 0.3048 m, otherwise the results will be dimensionally inconsistent.

Impact velocity is the total speed when the projectile lands, found by combining the horizontal and vertical velocity components with the Pythagorean theorem: v = √(vₓ² + v_y²).

The horizontal component stays constant at vₓ = v₀·cos(θ), while the vertical component at landing is v_y = v₀·sin(θ) − g·t.

For an object simply dropped from height h (θ = 0, v₀ = 0), impact speed simplifies to v = √(2gh): dropped from 45 m, v = √(2 × 9.81 × 45) = √882.9 ≈ 29.7 m/s.

Khan Academy explains that in a vacuum, launch speed equals landing speed when launch and landing heights are identical, since energy is conserved.

In projectile motion the horizontal and vertical directions are analyzed separately because gravity acts only vertically.

The horizontal velocity vₓ = v₀·cos(θ) is constant (no horizontal force in a vacuum), so horizontal position grows linearly: x = v₀·cos(θ)·t. Vertically, gravity produces constant downward acceleration g, giving y = v₀·sin(θ)·t − ½g·t².

This independence, first articulated by Galileo and detailed by Encyclopaedia Britannica, means a bullet fired horizontally and a bullet dropped from the same height hit the ground simultaneously.

HyperPhysics treats the two components as two one-dimensional kinematics problems whose only shared variable is time t.

When a projectile launches from an elevated point, time of flight comes from solving the vertical position equation 0 = h + v₀·sin(θ)·t − ½g·t² with the quadratic formula: t = [v₀·sin(θ) + √((v₀·sin(θ))² + 2gh)] / g. Here h is the launch height above the landing surface.

For launch from ground level (h = 0) this reduces to the standard T = 2·v₀·sin(θ)/g.

Because the discriminant grows with h, added height always increases both flight time and horizontal range. Only the positive root is physically meaningful, since the projectile lands after launch.

Khan Academy demonstrates this quadratic approach for cliff and rooftop launch problems.

Projectile equations model many real activities.

• In basketball a free throw released around 7 m/s near 52° follows a parabola into a rim 3.05 m high.
• In ballistics and artillery, gunners set elevation angles using range tables derived from R = v₀²·sin(2θ)/g, then correct for air drag, wind, and Earth's rotation (the Coriolis effect).
• Civil and mechanical engineers apply the same kinematics to water-fountain jet design, fire-hose reach, and the launch of packages from conveyor systems.
• Aerospace engineers use projectile fundamentals as the vacuum baseline before adding aerodynamic drag and lift.

Encyclopaedia Britannica notes that these idealized parabolic results remain the essential first approximation across sports science and engineering.

The most frequent errors are mixing up angle modes and units.

• Calculators default to radians, so sin(45°) must use degree mode or the radian value 0.7854, not 45.
• Another mistake is forgetting that only the vertical component feels gravity — the horizontal velocity vₓ = v₀·cos(θ) never changes in a vacuum.
• Students also misuse T = 2v₀sin(θ)/g when the launch and landing heights differ; that shortcut only holds for flat ground.
• Sign conventions matter too: treat downward acceleration as −g consistently.
• Finally, do not equate maximum height with half the range, and remember this model ignores air resistance, per NIST and HyperPhysics idealized-motion assumptions.

Because g appears in the denominator of every projectile formula, weaker gravity produces a longer, higher, slower flight. Range scales inversely with g: R = v₀²·sin(2θ)/g.

Using NIST-style standard values:

• Earth's g is 9.81 m/s²,
• the Moon's is about 1.62 m/s²,
• and Mars's is about 3.72 m/s².

A projectile launched at 20 m/s at 45° travels roughly 40.8 m on Earth but about 246.9 m on the Moon (40.8 × 9.81 / 1.62) — about six times farther. Time of flight and maximum height rise by the same factor.

This is why the calculator lets you edit g to model lunar, Martian, or laboratory-adjusted conditions.