Math & Scientific

Projectile Motion with Drag: Linear and Quadratic Models

Set up velocity-dependent drag equations, check a linear-drag position example, and distinguish numerical trajectory work from the local no-drag calculator.

By Uddhav Gopani, FounderPublished 25 September 2026Reviewed 29 September 2026

Projectile Motion with Drag: Linear and Quadratic Models

Choose a drag model before seeking a range formula

Air resistance changes as a projectile's velocity changes, so the familiar no-drag range formula cannot generally be corrected by subtracting one fixed drag force. Specify mass, initial position and velocity, gravitational acceleration, air motion and a drag law. Two common mathematical models are linear drag F_D = −bv and quadratic drag F_D = −c|v|v in still air. Their constants have different units and are not interchangeable. A model appropriate to one object and flow regime may be inappropriate to another, even when the initial speeds happen to match.

Write the linear-drag differential equations

For constant mass and linear drag, define β = b/m, with units s⁻¹. With x horizontal, y upward, gravity magnitude g and no wind, the equations are dv_x/dt = −βv_x and dv_y/dt = −g − βv_y. Position changes according to dx/dt = v_x and dy/dt = v_y. These equations show why vertical acceleration is not a constant −g: drag adds a velocity-dependent term. On descent, the drag term opposes downward motion. Replacing v_y with its unsigned magnitude would lose that required direction change.

Use the closed-form linear solution under its assumptions

With initial coordinates zero and initial velocity components u_x and u_y, the linear model gives x(t) = u_x(1 − e^(−βt))/β and y(t) = (u_y + g/β)(1 − e^(−βt))/β − gt/β. These expressions assume constant positive β and constant g. They satisfy x(0) = y(0) = 0; differentiating gives the velocity equations above. They are not the solution for quadratic drag. As drag tends to zero, their limits recover x = u_xt and y = u_yt − ½gt², although directly dividing by β = 0 is undefined.

Check a numerical position at one second

Take the illustrative values u_x = 10 m/s, u_y = 10 m/s, β = 0.20 s⁻¹ and g = 10 m/s². At t = 1 s, e^(−0.2) is approximately 0.818730753. Horizontal position is 50(1 − e^(−0.2)) ≈ 9.06346 m, while vertical position is 300(1 − e^(−0.2)) − 50 ≈ 4.38077 m. Without drag, the same initial components and chosen g give 10 m horizontally and 5 m vertically. These are model comparisons at one time, not a computed landing range or a prediction for an unspecified real object.

Form the coupled quadratic-drag equations

For the common aerodynamic model c = ½ρC_DA, where ρ is air density, C_D is a dimensionless drag coefficient and A is the matching reference area. Let γ = c/m and speed v = sqrt(v_x² + v_y²). Then dv_x/dt = −γv v_x and dv_y/dt = −g − γv v_y. The components are coupled through the total speed. Using −γv_x² and −γv_y² separately gives both an incorrect magnitude relationship and incorrect signs for negative components. With wind, use air-relative velocity in the drag term while retaining ground-relative velocity for position.

Check one quadratic-drag acceleration state

For a defined example with ρ = 1.2 kg/m³, C_D = 0.5, A = 0.010 m² and m = 0.20 kg, c = 0.003 kg/m and γ = 0.015 m⁻¹. At v_x = v_y = 10 m/s, speed is sqrt(200) m/s. Using g = 10 m/s² gives a_x ≈ −2.12132 m/s² and a_y ≈ −12.12132 m/s². This evaluates one instantaneous state. A trajectory requires repeatedly updating both velocity components and position, with an integration method, a step-size check and a defined ground-crossing condition.

Match the local calculator to a supported step

The linked Physics Calculator can evaluate expressions such as 50*(1-e^(-0.2)) or -0.015*sqrt(200)*10. It does not integrate a drag trajectory or estimate an object's drag coefficient. The separate Projectile Motion Calculator explicitly uses no air resistance, level launch and landing height, and g = 9.81 m/s²; it cannot validate the drag examples above. For a numerical trajectory, compare progressively smaller integration steps, handle the landing event between samples and document the chosen drag law. Agreement with a no-drag limit is a useful check, but it does not establish the accuracy of an unmeasured aerodynamic model.