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Physics Tools · Updated October 21, 2026 · Reviewed by FormatFlow Editorial

Projectile Motion with Drag Calculator

You can model a projectile with quadratic air drag, numerically integrating the path to get range, flight time, maximum height and impact speed, and comparing each result with the ideal vacuum trajectory.

Runs in your browserFree · No sign-up · No watermark
v₀, θ, k → rangeRuns in your browser
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Everything runs locally in this tab. No file is uploaded while the tool loads.

This tool runs in your tab. Nothing is uploaded and no copy is stored.

How files are handled

At a glance

Works with
v₀, θ, k → range
Processing
In your browser
Price
Free, no account
Updated
October 21, 2026
All Physics Tools

About Projectile Motion with Drag Calculator

Air resistance changes a projectile's flight in ways the vacuum model cannot capture: the path becomes asymmetric, the descent is steeper and the range is shorter. For fast or light objects the difference is large enough that an ideal calculation is only a starting point.

This calculator integrates a quadratic drag model step by step, then compares the result with the analytic vacuum trajectory. You enter the launch conditions, mass and drag coefficient, and get range, flight time, maximum height, impact speed and a plot of both paths.

What this tool does

The details that matter when you use Projectile Motion with Drag Calculator for real work.

  • Quadratic drag model

    Drag magnitude k·v² opposes velocity, so the acceleration is −g·ŷ − (k/m)·|v|·v.

  • Numerical integration

    Fourth-order Runge–Kutta advances the flight until the projectile returns to the ground, with the landing point refined by interpolation.

  • Vacuum comparison

    The same launch is solved analytically without drag, so range, flight time and impact speed can be compared directly.

  • Trajectory plot

    The drag path is drawn solid and the vacuum path dashed, making the flattened descent visible.

  • Drag parameter explained

    k is the lumped parameter ½ρC_dA, shown on the page; k = 0 switches to the vacuum case.

  • Editable inputs

    Initial height, gravity and mass units can all be changed, so textbook values and other worlds are supported.

How to use Projectile Motion with Drag Calculator

No account, no installation — the whole job happens in this browser tab.
  1. 01

    Enter the launch conditions

    Set the initial speed in m/s, the launch angle in degrees and the release height in meters.

  2. 02

    Set gravity and mass

    Keep 9.81 m/s² for Earth or change it, and enter the projectile mass in kilograms or grams.

  3. 03

    Enter the drag coefficient k

    Use k = ½ρC_dA in kg/m, or enter 0 to compare with the vacuum model.

  4. 04

    Read the drag results

    The headline gives the drag range, while the cards show flight time, maximum height, impact speed, vacuum values and the reduction percentage.

  5. 05

    Compare the plotted paths

    The solid drag curve and dashed vacuum curve show how drag flattens the flight before Reset clears the inputs.

How this tool works

Works with
v₀, θ, k → range
Processing
Client-side, in your browser. Nothing is uploaded.
Requirements
Chrome, Edge, Firefox or Safari · JavaScript enabled

Limitations worth knowing

  • Assumes constant air density, a constant quadratic drag coefficient and no wind, spin or lift.
  • Models a single plane over flat ground; vertical variation in density and curved trajectories are not included.
  • The value of k is the main uncertainty, since real drag depends on speed, shape and orientation.

Privacy and file handling

Launch conditions, mass and drag coefficient are processed entirely in your browser. Nothing is uploaded, stored or logged, and the trajectory is computed locally.

Read the full privacy policy

Projectile Motion with Drag Calculator FAQ

Short, direct answers to the questions people ask before using this tool.
What drag model does the calculator use?
It uses quadratic drag, where the force magnitude is k·v² and the force points opposite to the velocity vector. That gives an acceleration of −g·ŷ − (k/m)·|v|·v, which is the standard model for fast projectiles where inertial drag dominates.
What is the drag coefficient k and how do I estimate it?
k is the lumped parameter ½ρC_dA, with air density ρ, drag coefficient C_d and cross-sectional area A. For a sphere of radius r, A is πr² and C_d is roughly 0.47; at sea-level density near 1.2 kg/m³, a 5 cm radius gives k of about 0.0022 kg/m.
What happens when k is zero?
A zero drag coefficient removes the drag term entirely, so the numerical trajectory matches the analytic vacuum path. This is useful for checking the integrator and for comparing the two models under identical launch conditions.
Why does drag reduce the range?
Drag opposes motion throughout the flight and removes mechanical energy, so the projectile cannot cover the same horizontal distance. The effect grows with speed and with the drag-to-mass ratio k/m, which is why light, fast objects are affected much more than heavy ones.
Why does the flight time change with drag?
Drag alters the vertical velocity as well as the horizontal one, slowing the rise and limiting the fall speed. The projectile often lands sooner than in vacuum, though the exact change depends on the launch angle and the drag-to-mass ratio.
Why is the trajectory asymmetric with drag?
Drag grows with speed, and the projectile is slowest near the apex and fastest during the descent. The descent therefore experiences more drag than the climb in relative terms, producing a steeper fall and an apex that is not at the midpoint of the flight.
Does the calculator model wind, spin or lift?
No. It assumes still air, no Magnus lift from spin, no wind and flat ground. Those effects can change a real trajectory substantially, so the model is best used for understanding the drag correction rather than predicting a specific real launch.
How accurate is the numerical integration?
The step size is chosen small relative to the drag relaxation time and the method is fourth-order Runge–Kutta, so numerical error is normally small. The dominant uncertainty is the model input k, which varies with speed, air density and the projectile's orientation.

Background reading

Format guides that explain the trade-offs behind Projectile Motion with Drag Calculator.