Kinematics - Velocity, force of gravity, terminal speed

[Sonia22]

Hi everyone,
I would really appreciate any advice on how to go about solving the following problem. I'm supposed to investigate it in a report.

Homework Statement

A paratrooper who jumps out of an aircraft moving in horizontal flight initially has the same horizontal velocity as the aircraft. However, immediately, forces begin to change this. The drag force begins to slow him down, but at the same time the force of gravity tends to speed him up. It has been claimed that for any specified launching speed, the speed of the paratrooper will pass through a minimum value less than either the launching speed or the terminal speed. Obviously, if he is high enough, the paratrooper should wait for the minimum speed to lessen the shock when the parachute opens. Investigate this claim.

Homework Equations

d=Vi(t) + 1/2a(t^2), v_term = (mg/k)^(1/3)

The Attempt at a Solution

I'm not completely sure how to approach this question, but I started by looking at the equations for kinematics and terminal velocity. btw I'm in Grade 12, if that helps.

 

[haruspex]

You quote the expression for terminal velocity. I assume you know from that the equation for the drag force.
Can you write out (and post) the differential equations? (Since you don't care about position, it's just a pair of first order equations.) Pretty sure there's no closed form of solution, though.
Several possible approaches:
- Write the condition for a minimum speed. Maybe can show it is less than terminal speed.
- It does say "any specified launching speed", so the claim could be disproved by finding some launch speed for which it doesn't work.
- Consider large t, so the horizontal speed is small and the difference between vertical speed and terminal speed is small. Making some approximations based on that, solve the differential equations and show that the speed is increasing. However, that would only establish the claim for the case where the launch speed is at least equal to the terminal speed.

On the other hand, all of these seem a bit advanced for grade 12, so maybe I'm missing something simpler.