Lessons · freefall

Falling with air resistance

Regime 2 — calculus does what algebra cannot. The acceleration isn't constant, so the algebra formulas don't apply; calculus is the only road in, and SymPy proves the closed form solves the equation of motion.

Regime 2 · calculus does more Math machine-derived & checked by SymPy 3 modeling assumptions (author-asserted)

An object is dropped from rest and feels air resistance proportional to its speed (m = 1 kg, drag b = 0.5 kg/s). Does it keep speeding up forever — and why can't the algebra formulas answer that?

Terminal velocity

mg/b

-20 m/s

Time constant

\tau = m/b

2 s

Newton's second law with linear drag

m\,v' = mg - b\,v \;\;\Longrightarrow\;\; v' = g - \tfrac{v}{\tau},\quad \tau = \tfrac{m}{b}

Drag opposes motion and grows with speed, so the net force — and the acceleration — shrink as the object speeds up. This is a differential equation, not algebra.

back-substitute $v(t)$ into $v' = g - v/\tau$ ; check the initial condition; check $dx/dt = v$ ; check $\lim v = g\tau$

✓ The closed form solves the equation of motion $v' = g - v/\tau$ . [structural]
✓ It matches the initial velocity $v(0) = v_0$ . [structural]
✓ Position is the integral of velocity ( $dx/dt = v$ ). [structural]
✓ The terminal velocity falls out: $v \to g\tau = mg/b$ as $t \to \infty$ . [structural]

Dimensional homogeneity: checked by SymPy (holds).

✗ Common misconception: “A falling object keeps accelerating until it lands.”

With air resistance the acceleration decays toward zero as drag grows to cancel gravity, so the object approaches a constant terminal velocity. On the a-t graph, $a \to 0$ — the opposite of constant-gravity free fall, where $a$ stays at $-10$ the whole way.

Modeling assumptions — author-asserted, disclosed not discharged

Air resistance is linear in speed (drag = -b·v). Real drag is often closer to quadratic; the linear model keeps the closed form clean.
g = -10 m/s^2, a clean-arithmetic simplification of -9.81 m/s^2.
The object is a point mass.

The stacked graph, fully annotated

A static rendering (Matplotlib) at the default parameters — the interactive version is in the Graph tab above.

Formulas used

Hover a formula to preview its reference entry; click to open it in the reference (or the concept graph):