Example
\(v_0 = 30, \theta = 45^{\circ}, h_0 = 0, g = 9.8 \rightarrow R \approx 91.84\,\text{m}\)
Projectile Motion Calculator
Calculate trajectory, range, maximum height, and time of flight for projectile motion with step-by-step solutions.
Explore the trajectory
Drag launch velocity, angle, initial height, and gravity. Watch how each parameter reshapes the parabolic path.
Quick presets
30 m/s
1100
45 °
090
0.0 m
0.050.0
9.80 m/s²
1.0025.00
Predict what will happen
When the launch is from ground level, which angle gives the longest range?
Try 30°, 45°, and 60° while keeping v₀, h₀, and g fixed.
Notice
Horizontal motion is constant velocity (no horizontal force after launch); vertical motion is constant acceleration −g. The two are independent and combine into the parabola you see.
Common mistake
Range = v₀² sin(2θ) / g only holds when h₀ = 0. With a non-zero launch height, you must solve the quadratic for time of flight first, then multiply by v₀ₓ.
Why it works
Decompose v₀ into v₀ₓ = v₀ cos θ and v₀ᵧ = v₀ sin θ. Each component evolves independently under its own equation — the trajectory is the curve traced by (x(t), y(t)).
Results
Final Answer
Range = 91.84 m · Max height = 22.96 m · Flight time = 4.33 s
Velocity components
\(v_{0x} = v_0\cos\theta = 21.21\,\text{m/s}\) · \(v_{0y} = v_0\sin\theta = 21.21\,\text{m/s}\)
Theory & Formula
Projectile motion is the motion of an object thrown or projected into the air, subject only to gravity. It follows a parabolic trajectory.
Key Equations: \(x(t) = v_0\cos\theta \cdot t\), \(y(t) = h_0 + v_0\sin\theta \cdot t - \tfrac{1}{2}g t^2\)
For maximum range with no initial height, the optimal launch angle is 45°.
\(R = \frac{v_0^2 \sin(2\theta)}{g},\quad h_{\max} = h_0 + \frac{v_0^2 \sin^2\theta}{2g}\)
Worked Examples
External educational resource
Explore projectile motion in PhET
Open PhET's Projectile Motion simulation to vary mass, drag, launch height and angle, and see the trajectory respond live.
PhET Interactive Simulations, University of Colorado Boulder