02 / CHAPTER 08
Drag and terminal speed
Compare linear and quadratic drag and identify a terminal regime.
ANCHOR EQUATION
Linear drag
In symbols: Fd = −bv
At low speed, drag is often modeled as proportional to velocity.
Drag creates a speed selected by force balance
Drag opposes motion relative to a fluid. At low Reynolds number it is often proportional to speed; at higher speed a bluff body usually experiences drag proportional to v². Terminal speed is not a new force—it is the speed where forces balance.
Set the acceleration to zero
For vertical fall with downward positive and linear drag b v upward, m dv/dt = mg − bv. At terminal speed dv/dt = 0, so mg = b v_t and v_t = mg/b. With quadratic drag c v², the same balance gives mg = c v_t² and v_t = √(mg/c). The drag law changes the scaling, not the balance method.
A 0.20 kg object with b = 0.50 kg/s has v_t = 0.20(9.81)/0.50 = 3.92 m/s in the linear model. Doubling the mass doubles terminal speed here, whereas in the quadratic model it increases speed by √2.
Terminal does not mean stationary
At terminal speed the object still moves and drag still acts; net force is zero. If density, orientation, or fluid speed changes, the terminal speed changes too. Near the ground, a falling object may not have reached the terminal regime yet.
Work it: For m = 0.10 kg and quadratic coefficient c = 0.020 kg/m, calculate terminal speed. Compare the drag force at half that speed with the weight and predict the sign of acceleration.