projectile motion calculator

Enter launch values to calculate trajectory characteristics (no air resistance).

Initial speed, v₀ (m/s)

Launch angle, θ (degrees)

Initial height, h₀ (m)

Gravity, g (m/s²)

Optional: Evaluate position at time t (seconds)

Results

Horizontal velocity (v_x): 0 m/s
Vertical velocity (v_y): 0 m/s
Time of flight: 0 s
Maximum height: 0 m
Time to peak: 0 s
Horizontal range: 0 m
Impact vertical velocity: 0 m/s
Impact speed: 0 m/s

What this projectile motion calculator does

This tool computes key values for a launched object moving under constant gravity. You provide initial speed, launch angle, and starting height, and the calculator returns the most useful trajectory numbers: total flight time, maximum height, and horizontal range.

It is ideal for quick physics checks, engineering estimates, and classroom demonstrations. The model assumes motion in a vacuum-like setting (no drag, no wind, and constant gravity), which is the standard introductory projectile model.

Inputs and outputs explained

Inputs

Initial speed (m/s): Launch speed at t = 0.
Launch angle (degrees): Measured from the horizontal. Positive angles launch upward.
Initial height (m): Vertical starting point above the ground.
Gravity (m/s²): Downward acceleration. Use 9.81 for Earth near sea level.

Outputs

v_x and v_y: Initial horizontal and vertical velocity components.
Time of flight: Total time until the projectile hits y = 0.
Maximum height: Peak vertical position reached.
Range: Horizontal displacement at impact.
Impact speed: Speed magnitude just before landing.

Core equations used

The calculator uses the standard constant-acceleration equations:

v_x = v_0 cos(θ) v_y = v_0 sin(θ) x(t) = v_x t y(t) = h_0 + v_y t - (1/2) g t² Time of flight from y(t)=0: t_f = (v_y + √(v_y² + 2gh_0)) / g

From these, we compute range as x(t_f), and if the projectile launches upward (v_y > 0), peak height as h₀ + v_y²/(2g).

Worked example

Suppose a ball is launched at 30 m/s and 45° from ground level with Earth gravity. This gives equal horizontal and vertical starting components, about 21.21 m/s each. The tool then estimates flight time, range, and peak height in a fraction of a second.

In this symmetric case (launch and landing at the same height), trajectory is easy to visualize: rise to a peak, then descend with increasing downward speed. Changing only angle while keeping speed fixed shows the familiar tradeoff between height and range.

How to use this calculator effectively

Use consistent units (m, s, m/s).
Set gravity for your context (Earth, Moon, or another planet).
Use the optional time query field to inspect position at a specific moment.
For downward launches, try negative angles to model throws below the horizontal.

Important: Real projectiles in air experience drag and wind. This calculator intentionally ignores those effects. Use it for baseline physics, not final safety-critical design.

Common mistakes to avoid

1) Mixing degrees and radians

The input angle is in degrees. Internally, the script converts to radians for trigonometric functions.

2) Entering gravity as a negative number

Gravity should be entered as a positive magnitude (for example, 9.81). Direction is already handled by the equations.

3) Expecting perfect real-world agreement

For long distances, light objects, or high speeds, air resistance significantly changes results.

Use cases

Physics homework and concept checks
Sports trajectory exploration (e.g., balls, throws)
Basic robotics and simulation prototyping
Quick educational demonstrations in STEM classes

Final thoughts

A projectile motion calculator is a practical way to connect equations with intuition. By experimenting with speed, angle, height, and gravity, you can immediately see how each variable shapes a trajectory. It is one of the fastest ways to build solid understanding of 2D kinematics.