Projectile Motion Calculator
Calculate the range, maximum height, and flight time of a projectile launched at an angle. Accounts for initial height above ground. See also our Kinetic Energy Calculator and Force Calculator.
How to Calculate Projectile Motion
Projectile motion is the motion of an object launched into the air, subject only to gravity and its initial velocity. This is one of the most studied problems in classical mechanics, first analyzed correctly by Galileo Galilei in the early 1600s. The key insight is that horizontal and vertical motions are independent — horizontal velocity remains constant (no air resistance), while vertical velocity changes due to gravity.
To solve a projectile motion problem, decompose the initial velocity into horizontal (v₀cos θ) and vertical (v₀sin θ) components. The horizontal position is x = v₀cos(θ)×t (constant velocity). The vertical position is y = h₀ + v₀sin(θ)×t - ½gt² (uniformly accelerated motion). The projectile lands when y = 0, which gives the total flight time, and substituting back gives the range.
For launch from ground level (h₀ = 0), the range is maximized at 45° launch angle. Complementary angles (like 30° and 60°) give the same range but different trajectories — the lower angle gives a flatter, faster path while the higher angle gives a higher, slower arc. When launched from an elevated position, the optimal angle is less than 45° because the projectile has extra time to travel horizontally during its descent.
Projectile Motion Formulas
Velocity Components:
v₀ₓ = v₀ × cos(θ) (horizontal, constant)
v₀ᵧ = v₀ × sin(θ) (vertical, initial)
Position Equations:
x(t) = v₀ₓ × t
y(t) = h₀ + v₀ᵧ × t - ½gt²
Range (from ground level, h₀=0):
R = v₀² × sin(2θ) / g
Maximum Height:
H = h₀ + v₀ᵧ² / (2g) = h₀ + v₀²sin²(θ) / (2g)
Time of Flight (from ground, h₀=0):
T = 2v₀sin(θ) / g
Time of Flight (with initial height):
T = [v₀ᵧ + √(v₀ᵧ² + 2gh₀)] / g
Trajectory Equation:
y = x×tan(θ) - gx²/(2v₀²cos²θ) + h₀
Example Calculation
A ball is launched at 20 m/s at 45° from ground level. Calculate range, max height, and flight time:
Given: v₀ = 20 m/s, θ = 45°, h₀ = 0 m, g = 9.81 m/s²
v₀ₓ = 20 × cos(45°) = 20 × 0.7071 = 14.142 m/s
v₀ᵧ = 20 × sin(45°) = 20 × 0.7071 = 14.142 m/s
Range: R = v₀²sin(2θ)/g = 400×sin(90°)/9.81 = 400/9.81 = 40.77 m
Max Height: H = v₀²sin²(θ)/(2g) = 400×0.5/19.62 = 10.19 m
Flight Time: T = 2v₀sin(θ)/g = 2×20×0.7071/9.81 = 2.88 s
Time to apex: t_apex = v₀sin(θ)/g = 14.142/9.81 = 1.44 s
Verify: Range = v₀ₓ × T = 14.142 × 2.88 = 40.77 m ✓
Projectile Motion Reference Table
| v₀ (m/s) | Angle (°) | h₀ (m) | Range (m) | Max H (m) | Time (s) |
|---|---|---|---|---|---|
| 10 | 30° | 0 | 8.83 | 1.27 | 1.02 |
| 10 | 45° | 0 | 10.19 | 2.55 | 1.44 |
| 10 | 60° | 0 | 8.83 | 3.82 | 1.77 |
| 15 | 45° | 0 | 22.94 | 5.74 | 2.16 |
| 20 | 30° | 0 | 35.31 | 5.10 | 2.04 |
| 20 | 45° | 0 | 40.77 | 10.19 | 2.88 |
| 20 | 60° | 0 | 35.31 | 15.29 | 3.53 |
| 20 | 45° | 5 | 44.95 | 15.19 | 3.18 |
| 30 | 45° | 0 | 91.74 | 22.94 | 4.33 |
| 50 | 45° | 0 | 254.84 | 63.71 | 7.21 |
| 100 | 30° | 0 | 882.79 | 127.42 | 10.19 |
| 100 | 45° | 0 | 1019.37 | 254.84 | 14.41 |
Frequently Asked Questions
What angle gives maximum range?
For launch from ground level (h₀ = 0) without air resistance, 45° gives maximum range. This is because range = v₀²sin(2θ)/g, and sin(2θ) is maximized when 2θ = 90°, i.e., θ = 45°. With air resistance, the optimal angle is typically 30-40° depending on the object's drag coefficient. When launching from an elevated position, the optimal angle is less than 45°.
Why do complementary angles give the same range?
Because sin(2θ) = sin(180° - 2θ). For example, sin(2×30°) = sin(60°) = sin(2×60°) = sin(120°) = sin(60°). So 30° and 60° give the same range, as do 20° and 70°, etc. The lower angle produces a flatter trajectory with shorter flight time, while the higher angle produces a higher arc with longer flight time. Both cover the same horizontal distance.
How does air resistance affect projectile motion?
Air resistance (drag) reduces both range and maximum height compared to the ideal case. It acts opposite to the velocity vector, so it decelerates the projectile in both horizontal and vertical directions. The trajectory becomes asymmetric — the descending portion is steeper than the ascending portion. For high-speed projectiles (bullets, baseballs), air resistance significantly reduces range. The drag force is proportional to v² for most practical cases.
What assumptions does this calculator make?
This calculator assumes: no air resistance, constant gravitational acceleration (flat Earth approximation), no wind, no spin effects (Magnus force), and the projectile is a point mass. These assumptions are reasonable for short-range, low-speed projectiles (throwing a ball across a field) but break down for long-range artillery, high-speed bullets, spinning balls (curveballs), and very high altitudes where g varies.
How do I find the velocity at any point in the trajectory?
The horizontal velocity is constant: vₓ = v₀cos(θ). The vertical velocity changes: vᵧ = v₀sin(θ) - gt. The speed at any time is: v = √(vₓ² + vᵧ²). At the apex, vᵧ = 0, so speed = vₓ = v₀cos(θ). At impact, use the total flight time to find vᵧ, then calculate the resultant speed. The impact speed equals the launch speed when h₀ = 0 (energy conservation).
What are real-world applications of projectile motion?
Sports (basketball shots, golf drives, javelin throws), military ballistics (artillery, missiles), engineering (water fountains, irrigation sprinklers), forensics (bullet trajectory analysis), space (orbital insertion burns), entertainment (fireworks, stunt jumps), and agriculture (crop dusting spray patterns). Understanding projectile motion helps optimize launch parameters for maximum range, accuracy, or specific landing conditions.
The Physics of Projectile Motion
The beauty of projectile motion lies in the independence of horizontal and vertical components. Gravity only affects the vertical motion — it pulls the projectile downward at 9.81 m/s² regardless of horizontal speed. A bullet fired horizontally and a bullet dropped from the same height hit the ground at the same time (ignoring air resistance). This independence allows us to solve complex 2D problems as two separate 1D problems, making the mathematics tractable.
Beyond Ideal Projectile Motion
Real projectiles experience air resistance (drag force proportional to v²), the Magnus effect (spinning objects curve due to pressure differences), the Coriolis effect (long-range projectiles deflect due to Earth's rotation), and varying gravity (for very high trajectories). Military ballistics computers account for all these factors plus wind, temperature, humidity, and barrel wear. Sports scientists use these effects intentionally — a curveball in baseball uses the Magnus effect, and a golf ball's dimples reduce drag while creating lift through backspin.
Projectile Motion in Sports
In basketball, the optimal shooting angle is typically 45-55° depending on release height and distance. In soccer, free kicks can reach 30+ m/s with significant spin causing curved trajectories. In golf, drives launch at 10-15° with backspin creating lift that extends range beyond the vacuum prediction. In javelin, the optimal release angle is about 35° due to the aerodynamic properties of the javelin. Understanding these principles helps athletes and coaches optimize technique for maximum performance.