The Drag Force and Terminal Speed — Question 4

Question 4

Acceleration data reveal a quadratic drag coefficient

A 2 kg body descends in still fluid under gravity and quadratic drag only. Two exact measurements are $(v,a)=(2,9.2)$ and $(6,2.8)$ in SI units, with downward acceleration positive. Treat local g and c in $D=cv^2$ as unknown.

1. Use the measurements to find g and c.

2. Infer terminal speed.

3. Predict acceleration at 8 m/s and explain which plot would test the model using additional measurements.

Question 4 - Solution

Acceleration data reveal a quadratic drag coefficient

(a) Subtract the force equations.

Use $a=g-(c/m)v^2$ with $m=2$ kg: $$9.2=g-2c,\qquad2.8=g-18c\quad\text{(SI numerical values)}.$$ Subtracting gives $-6.4=-16c$, so $$\boxed{c=.40\ \mathrm{kg/m}},\qquad\boxed{g=9.2+.20(4)=10\ \mathrm{m/s^2}}.$$

(b) Find the zero-acceleration speed.

Set the fitted acceleration to zero: $$0=10-.20v_t^2\Longrightarrow\boxed{v_t=\sqrt{50}=5\sqrt2=7.071\ \mathrm{m/s}}.$$

(c) Extrapolate and propose a diagnostic.

At the proposed speed, $$\boxed{a(8)=10-.20(8)^2=-2.8\ \mathrm{m/s^2}}.$$ The body is descending but slowing. Additional data should satisfy $a=g-(c/m)v^2$: a straight $a$-versus-$v^2$ plot with intercept $10$ m/s$^2$ and slope $-.20$ m$^{-1}$.