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⚡ General Physics & Classical Mechanics

Projectile Motion Calculator

Simulate 2D projectile kinematics with real-time trajectory plotting. Calculate total range, flight time, maximum apex height, and instantaneous velocity vectors.

Presets:

Launch Parameters

m/s
45°
meters
m/s²
Total Range (R)
63.71 m
Horizontal distance
Max Height (H)
15.93 m
Peak above ground
Flight Time (t)
3.60 s
Total hang time
Impact Speed
25.00 m/s
Final magnitude

🎯 Real-Time Flight Trajectory

y(x) parabolic arc

📝 Kinematics Step-by-Step Derivations

How to Use This Calculator

  1. Set the initial velocity v₀ and the launch angle θ — use the slider or type a value directly in the box.
  2. Optionally set a launch height y₀ and change the gravitational acceleration g (try the Moon or cliff presets).
  3. Read the total range, maximum height, flight time, and impact speed from the four metric cards — the velocity components are broken down in the derivation.
  4. Watch the trajectory canvas: the parabola, apex, and landing point redraw in real time as you edit any input.
  5. Read the step-by-step derivation below the canvas to see every formula evaluated with your own numbers.

Physics Behind Projectile Motion

Projectile motion is the two-dimensional motion of an object thrown or projected into the air, subject only to acceleration as a result of gravity. The horizontal and vertical motions are completely independent of each other:

  • Horizontal motion (x-axis): Zero acceleration (a_x = 0). Constant velocity v_x = v₀ cos(θ).
  • Vertical motion (y-axis): Constant downward acceleration (a_y = -g). Governed by SUVAT kinematic equations.

Core Governing Equations

Maximum Apex Height (H)

H = y₀ + (v₀ sin θ)² / (2g)

Calculated where vertical velocity component vy momentarily equals zero.

Horizontal Range (R)

R = v₀ cos(θ) × t_flight

For flat ground (y₀ = 0), simplifies to R = (v₀² sin 2θ) / g.

Assumes no air resistance and constant g; the range formula R = v₀²sin2θ/g applies when launch and landing heights are equal (y₀ = 0).

Common Questions & Answers

Projectile Motion Calculator FAQs

What launch angle produces the maximum range in projectile motion?
When launched from ground level (initial height y0 = 0) with no air resistance, a launch angle of 45 degrees yields the maximum horizontal range because sin(2θ) reaches its maximum value of 1 at 2(45°) = 90°.
Why is horizontal velocity constant in ideal projectile motion?
Assuming negligible air resistance, gravity acts solely in the vertical downward direction. Because there is no horizontal acceleration (ax = 0), the horizontal velocity vx remains constant throughout flight: vx = v0 cos(θ).

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Reviewed & updated: October 2026 Maintained by the OnlineResult Study Tools Team Report an issue
Sources & further reading:

Disclaimer: This calculator models idealized projectile motion (no air resistance, constant gravitational acceleration) for educational purposes only. Do not rely on it for safety-critical applications.

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