An example on a physics assignment asks this question:
NASA is preparing a probe to send to Mars. The probe weighs $40kg_f$ on Earth. As it approaches Mars, the gravitational field of Mars, which is given as $g_{mars}=3.711 m/s^2$, will pull it down. The question is, what should the diameter of the parachute be so that the probe touches the surface with a speed of $3m/s$?
Here's what's given:
- $A_{chute} = \pi \frac{D^2}{4}$
- $C_D$ (the parachute's drag coefficient) $=1.4$
- Density of Mars' atmosphere = $0.8167kg/m^3$
- Landing the probe on Mars at $3m/s$ is equivalent to dropping it from a height of $0.5 $ meters (sans parachute) on Earth.
- The problem involves a differential equation.
Here's where I'm stuck: I'm using the formula $ m \frac{dv}{dt} = \frac{1}{2} \rho_{air} \space C_d A \space v^2$ to solve for $A$, area. The problem is, that when the parachute is drifting down through the Martian atmosphere with the parachute deployed, it quickly meets its terminal velocity and $\frac{dv}{dt}=0$, which makes it impossible to solve for $A$. I can't find another way/formula to use to solve for $A$.