What drag model does this tool use?

It uses the quadratic drag model: F = 0.5 × ρ × C_d × A × v², where ρ is air density, C_d is the drag coefficient, A is the cross-sectional area, and v is the current speed. This is accurate for most macroscopic projectiles at typical velocities.

How accurate is the numerical integration?

The tool uses the Euler method with a 1-millisecond time step. For typical low-speed projectiles this gives results within ~0.1% of the analytical solution. For very high speeds or long flight times you may see small cumulative errors.

What is a typical drag coefficient?

A smooth sphere has C_d ≈ 0.47, a baseball ≈ 0.30–0.35, a golf ball ≈ 0.24 (with dimples), and a flat plate ≈ 1.17. Streamlined bodies can be 0.04 or lower.

Why is the drag range always shorter than the vacuum range?

Air drag continuously removes kinetic energy from the projectile. The horizontal deceleration reduces how far it travels, and the asymmetric trajectory (steeper descent) causes it to land sooner than the symmetric vacuum parabola.

Can I model a bullet or artillery shell?

Yes — enter the appropriate mass, diameter (to compute area as π r²), and C_d. Note that supersonic projectiles experience different drag regimes; this tool assumes subsonic, constant C_d.

Projectile Motion with Air Drag Calculator

Simulate and plot a projectile trajectory with realistic air drag using numerical integration — adjust mass, drag coefficient, and launch angle to see the effect.

Launch Parameters

Launch Speed (m/s)

Launch Angle (°)

Projectile Properties

Mass (kg)

Cross-section Area (m²)

Drag Coefficient C_d

Air Density ρ (kg/m³)

Quick Presets:

Trajectory Plot

With Drag Vacuum

Set parameters and click Simulate to see the trajectory

Summary

This tool numerically integrates the equations of motion for a projectile subject to quadratic air drag (F_drag = ½ · ρ · C_d · A · v²). It plots both the drag-affected and the ideal (vacuum) trajectories side by side so you can visualize how air resistance shortens range and reduces peak height. All computation runs in your browser — no data is sent to any server.

How it works

Enter the launch speed (m/s) and angle (degrees above horizontal).
Set the projectile mass (kg), cross-sectional area (m²), and drag coefficient (C_d).
Optionally adjust air density (default 1.225 kg/m³ at sea level).
Click "Simulate" — the tool solves the drag ODE with a 1 ms Euler step and draws the trajectory on a Canvas chart.
Key results (range, max height, flight time) are shown for both the drag and vacuum cases.

Use cases

Physics students verifying drag effects on ballistic range.
Engineers estimating projectile flight paths under aerodynamic drag.
Comparing ideal vacuum trajectories to real-world drag-affected paths.
Exploring how changes in drag coefficient alter flight distance.
Understanding the effect of launch angle on range with and without drag.
Sports science — modeling balls with known drag coefficients (baseball, soccer, golf).

Frequently Asked Questions

Last updated: 2026-05-23 · Reviewed by Nham Vu