2.7 Resistive Force

Drag forces depend on velocity

A resistive force (drag) opposes an object's motion through a fluid and grows with speed. At low speeds, drag is often modeled as linear: F = -bv. At higher speeds, it's often quadratic: F = -bv². In both cases the force direction is always opposite the velocity, so it acts like a velocity-dependent friction that changes as the object speeds up or slows down.

Terminal velocity

As a falling object speeds up, drag increases until it balances gravity (net force = 0). At that point acceleration = 0 and the object falls at constant terminal velocity, v_t. For linear drag, v_t = mg/b; for quadratic drag, v_t = √(mg/b). Terminal velocity is reached asymptotically — the object approaches it but technically never fully attains it.

Solving resistive force problems

Since drag depends on v, Newton's second law becomes a differential equation: m(dv/dt) = mg - bv (or -bv²). Solving requires separation of variables and integration, yielding v(t) that approaches terminal velocity exponentially. These problems test your ability to set up and solve first-order ODEs using calculus, not just algebra.

Key terms

Resistive force (drag)

A force opposing motion through a fluid, dependent on velocity; direction always opposite to velocity.

Linear drag model

F = -bv; used for slower speeds, where resistive force is proportional to velocity.

Quadratic drag model

F = -bv²; used for higher speeds, where resistive force is proportional to velocity squared.

Terminal velocity

The constant velocity reached when drag force balances gravitational force, resulting in zero net force and zero acceleration.

Why resistive force problems need calculus

Because acceleration depends on velocity, Newton's second law becomes a differential equation (m dv/dt = mg - bv) requiring integration to solve for v(t).

Try an AP-style question

An object falls with drag force F = −bv. What is its terminal velocity?

1. b/mg
2. √(mg/b)
3. mg/b
4. g/b