Choose a drag law by its regime

Let the object's velocity relative to the local undisturbed fluid be u. In an isotropic drag approximation, resistance points opposite u. Wind therefore matters even if a projectile's velocity relative to the ground is unchanged. A stationary object in flowing air can experience drag; a freely drifting object moving with uniform fluid has zero relative speed in this simplified model. Lift, buoyancy, added-mass effects, and history-dependent fluid forces are separate contributions. Omitting them is an assumption to evaluate for the apparatus, not a definition that every fluid interaction must fit the drag arrow.

Linear resistance uses a positive coefficient b with units kilograms per second. Quadratic resistance uses a positive coefficient c with units kilograms per metre, so multiplying it by speed squared produces newtons. The quadratic vector form contains speed times velocity, ensuring that it reverses direction when motion reverses. Writing a one-dimensional force as minus c times velocity squared for all signed velocities is wrong: it points in the same direction on both sides of zero. Express the vector or signed law first, and specialize to a chosen direction only after identifying the relevant branch.

Take downward positive for a dense falling object in still fluid and neglect buoyancy. Under linear drag the acceleration decreases as speed approaches the value at which drag balances weight. Solving the first-order velocity equation gives exponential relaxation with time constant mass divided by b. Terminal speed is approached asymptotically in this model; it is not a maximum allowed by Newtonian mechanics. An object released faster than terminal speed slows down toward it. Changing the initial velocity changes the transient while leaving the same equilibrium speed for fixed coefficients and constant gravity.

Quadratic drag produces a different terminal-speed scaling: doubling mass at fixed c increases terminal speed by the square root of two, whereas linear drag increases it by two at fixed b. A geometric change can also change the coefficient, so this comparison requires controlled shape and fluid conditions. If buoyancy is appreciable, replace weight by the net constant downward driving force before finding terminal speed. A nearly neutrally buoyant object cannot be modeled by simply carrying over the dense-object result. The equilibrium tests the complete force balance, including every retained constant force.

For a small sphere in creeping flow, Stokes resistance is linear with b equal to six pi times dynamic viscosity times radius. Its standard assumptions include a Newtonian fluid, no-slip surface, negligible inertial fluid effects, and boundaries sufficiently far away. Reynolds number compares fluid inertia with viscous effects using density, speed, characteristic size, and dynamic viscosity. Very small Reynolds number supports the creeping-flow limit. At larger Reynolds number a quadratic representation may be useful over a range where the drag coefficient is approximately constant, but that coefficient can vary strongly across transitions or with shape.

Measure a drag curve rather than deciding the law solely from the phrase low speed or high speed. Plot force against speed and against speed squared, inspect residuals, and check that the inferred coefficients remain consistent across repeated trials. A mixture of effects may require a combined or more detailed model. Even a good empirical curve does not automatically predict unsteady motion, rapid rotation, or flow near a wall. State the interval over which the fit was tested and propagate coefficient uncertainty into the predicted terminal speed. Modeling precision should not exceed the reliability of the chosen fluid approximation.