Projectile Motion and Kinematics Simulator
Model ideal projectile motion or one-dimensional constant acceleration with worked equations, graphs, vectors, and timeline controls.
Projectile motion and 1D kinematics
Two modes share one constant-acceleration model. Projectile mode animates an ideal launch with no air resistance and reports range, maximum height, time of flight, and the live velocity vector. One-dimensional mode plots position-time and velocity-time graphs for motion under constant acceleration.
- Set initial speed, launch angle, launch height, and gravity — with Earth, Moon, and Mars presets.
- Play, pause, and scrub the timeline, or slow it to 0.25× to read the motion at any instant.
- The grid is in real metres with equal scaling on both axes, so a steep launch looks steep.
Equations used
For a projectile launched at speed v₀ and angle θ from height h₀ under gravity g:
- Velocity components: vₓ = v₀·cos θ, v_y₀ = v₀·sin θ
- Position: x(t) = vₓ·t, y(t) = h₀ + v_y₀·t − ½·g·t²
- Time of flight: t = (v_y₀ + √(v_y₀² + 2·g·h₀)) / g
- Maximum height: h = h₀ + v_y₀² / (2·g)
- Range: R = vₓ · t
- Constant acceleration in 1D: v = v₀ + a·t, Δx = v₀·t + ½·a·t², v² = v₀² + 2·a·Δx
Frequently asked questions
- What launch angle gives the maximum range?
- From ground level with no air resistance, 45° gives the maximum range, because range is proportional to sin(2θ) and sin(2θ) peaks at θ = 45°. Launching from a height above the landing surface lowers the optimum angle below 45°, and real air resistance lowers it further.
- How do you calculate the time of flight of a projectile?
- Solve the vertical position equation for when the projectile returns to the ground: h₀ + v₀·sin θ·t − ½·g·t² = 0. The positive root is t = (v₀·sin θ + √((v₀·sin θ)² + 2·g·h₀)) / g. Launching from ground level (h₀ = 0) simplifies this to t = 2·v₀·sin θ / g.
- Does horizontal velocity change during projectile motion?
- Not in the ideal model. With no air resistance the only force acting is gravity, which is vertical, so the horizontal velocity component stays constant for the whole flight and only the vertical component changes. That is why the horizontal and vertical motions can be analysed separately.
- How does gravity change projectile motion on the Moon or Mars?
- Lower gravity means longer flights and greater range for the same launch. Lunar gravity is about 1.62 m/s² and Martian gravity about 3.71 m/s², compared with 9.81 m/s² on Earth. Since range is inversely proportional to g, the same launch travels roughly six times as far on the Moon and about 2.6 times as far on Mars.