Projectile Motion Calculator — Calculate Range, Height, and Time of Flight

The Physics of Projectile Motion

Projectile motion assumes:

• The only force acting is gravity (no air resistance)
• Gravity acts downward at g = 9.81 m/s² (on Earth)
• The horizontal and vertical components of motion are independent

The key insight is that a projectile's horizontal velocity remains constant (no horizontal force), while its vertical velocity changes at a constant rate of −g = −9.81 m/s² (deceleration upward, acceleration downward). The resulting path is a parabola.

Projectile Motion Equations

How to Use the Projectile Motion Calculator

1. Open the projectile motion calculator.
2. Enter the launch speed v₀ (m/s) — the initial speed of the projectile.
3. Enter the launch angle θ (degrees) — the angle above the horizontal.
4. Optionally, set the initial height if launching from above ground level.
5. Click Calculate. The tool returns range, maximum height, time of flight, and plots the trajectory.

Effect of Launch Angle on Range and Height

Worked Example: Football Kick

A football is kicked with initial speed 20 m/s at 45° to the horizontal. Find range, maximum height, and time of flight (ignoring air resistance).

Given: v₀ = 20 m/s, θ = 45°, g = 9.81 m/s²

• Range: R = v₀² sin(2θ) / g = 20² × sin(90°) / 9.81 = 400/9.81 ≈ 40.8 m
• Time of flight: T = 2v₀ sin(θ) / g = 2 × 20 × sin(45°) / 9.81 = 2 × 20 × 0.707 / 9.81 ≈ 2.89 s
• Maximum height: H = (v₀ sin θ)² / (2g) = (20 × 0.707)² / (2 × 9.81) = (14.14)² / 19.62 = 200/19.62 ≈ 10.2 m

Verify peak time: t_peak = T/2 = 2.89/2 ≈ 1.44 s. The football reaches maximum height after 1.44 seconds.

The Symmetry of Projectile Motion

For a projectile launched from and landing at the same height (horizontal ground):

• The time to maximum height equals the time from peak to landing: t_up = t_down = T/2
• The speed at landing equals the launch speed (kinetic energy is conserved)
• The trajectory is a perfect parabola, symmetric about the peak
• Complementary angles (θ and 90°−θ) give the same range: 30° and 60° both give the same horizontal range but different heights

Maximum Range and the 45° Rule

For a given launch speed v₀ on flat ground (without air resistance), range is maximised at θ = 45°. This can be derived from R = v₀² sin(2θ) / g — sin(2θ) is maximised when 2θ = 90°, i.e., θ = 45°.

The maximum range is: R_max = v₀²/g. For v₀ = 20 m/s: R_max = 400/9.81 ≈ 40.8 m.

However, 45° is only optimal without air resistance. With air resistance (the real world), the optimal angle for maximum range is lower — typically 30–40° for sports balls and bullets. Air resistance acts on the horizontal component of velocity, so a lower angle (which reduces horizontal flight time) minimises the drag penalty.

Projectile Motion with Initial Height

When a projectile is launched from height h above the landing point (e.g., thrown from a cliff or building), the symmetry is broken and the equations become:

• Time of flight: solve y(t) = h + v₀ sin(θ)t − ½gt² = 0 for t (quadratic formula required)
• Optimal launch angle for maximum range is less than 45° when h > 0

The calculator handles initial height — enter a non-zero initial height and the calculator uses the full quadratic solution.

Projectile Motion on Other Planets

The same equations apply on any body with a gravitational field — replace g = 9.81 m/s² with the local gravitational acceleration:

• Moon: g = 1.62 m/s² — projectiles travel ~6× further at the same launch conditions as on Earth
• Mars: g = 3.72 m/s² — projectiles travel ~2.6× further than on Earth
• Jupiter: g = 24.8 m/s² — projectiles travel ~0.4× as far as on Earth

The Apollo astronauts demonstrated this visually by dropping a hammer and feather simultaneously on the Moon — in the absence of air and with reduced gravity, both hit the ground at the same time (Galileo's prediction confirmed).

Real-World Applications

Sports science

Biomechanists use projectile motion to optimise launch conditions in shot put, javelin, long jump, and golf. Shot putters release at approximately 40° (not 45°) because the release height is above the landing height, shifting the optimum. Long jumpers produce forward velocity (not purely upward) to maximise the horizontal range while within their natural jumping capability.

Military ballistics

Artillery fire uses projectile motion to calculate firing elevation for a given range. Real artillery calculations include air resistance, wind, Earth's rotation (Coriolis effect), altitude, and variation in gravitational field — but the idealised projectile motion model provides the starting point.

Engineering design

Water fountain arcs, irrigation sprinkler trajectories, and industrial spray systems all use projectile motion principles to design nozzle angles and pressures for desired water distribution patterns.

Common Questions

Why is the maximum range at 45° and not some other angle?

At 45°, the vertical and horizontal components of launch velocity are equal (v₀/√2 each). Increasing the angle above 45° increases maximum height but reduces horizontal distance covered per unit time; decreasing below 45° does the opposite. The 45° balance point maximises the product of horizontal speed and time of flight that determines range.

Why doesn't air resistance matter in basic problems?

Introductory physics problems neglect air resistance to reveal the underlying gravitational physics clearly, with clean parabolic paths and symmetric equations. In reality, air resistance significantly affects real projectiles: a 100 m/s rifle bullet loses substantial speed over its path; a tennis ball's trajectory is strongly affected by air drag. Advanced ballistics and sports science use numerical methods with air resistance modelled explicitly.

Do both horizontal and vertical components of velocity change during flight?

No. In the absence of air resistance, the horizontal component is constant throughout flight — there is no horizontal force to change it. Only the vertical component changes (at rate −g = −9.81 m/s²). This independence of horizontal and vertical motion is the central insight that makes projectile motion problems tractable.