The question I'm having trouble with is:
A paratrooper steps out of an airplane at a height of 1000 ft and after 5 seconds opens her parachute. Her weight (with equipment) is 195 lbs. Let $y(t)$ denote her height above the ground after $t$ seconds. Assume the force due to air resistance is $0.005y'(t)^2 $ lbs in free fall and $0.6y'(t)^2$ lbs with the chute open.
- At what height does the chute open?
- How long does it take to reach the ground?
- At what velocity does she hit the ground?
So far I know that I need to use the equation $my''(t)=mg-ky'(t)$
I have $m=195/32$ and $ky'(t)$ a piecewise function depending on t (greater than or less than 5 sec). I'm wondering if I'm missing something because I don't have the initial height anywhere in my equation, and am confused on how to answer the first question without the initial height. I was also hoping someone could help me understand why the resistance in the prompt has $y'(t)^2$ and not $y'(t)$.
The initial height does not appear in the differential equation itself. After you solve a differential equation, the solutions will contain one or more integration constants, which can then be determined by requiring that initial conditions be satisfied.
In other words, the differential equation has a whole family of solutions, and your initial conditions, $y(0) = 1000$ and $y'(0) = 0$, will select a specific one among them.
Watch out for the units in the problem. The use of lbs as units for both mass and force is, in my opinion, a rather blatant abuse of units. Your writing $m = 195/32$ does do the trick, I guess, but I would have said that what is implied in the statement of the question is that we should take $m = 195$ and $g = -32$ (the minus to show that acceleration is down - that would be consistent with $y$ measuring "height", i.e. positive is up; in other words, also watch out for how you set up your forces in the diff equation with respect to directions up/down), and write the equation like this $$ \frac{m}{|g|}y''(t) = -m + ky'(t) $$ which is the same as what you get by taking $m = 195/32$ (at least as far as unit scaling is concerned - yours has the wrong signs in some places).