Question

James has just jumped out of an airplane. After he opens his parachute he experiences 2 forces: the constant force of gravity, and a wind drag that is proportional to his velocity. His height may therefore follow the equation: d²h/dt² = -9.8 - 2(dh/dt). As a second-order differential equation, this is not technically solvable by separation of variables. However, because the variable h appears only in its derivatives, we can turn this into a first-order equation, and solve that by separation.

a) Letting v = dh/dt, rewrite the above equation as a first-order differential equation in v.

b) Your equation in part a suggests that there is one velocity for which dv/dt = 0. What is this velocity?

c) Solve your equation using separation of variables. Your solution should contain an arbitrary constant: call it C₁.

d) Calculate lim t to infinite v(t) and use it to describe what is physically happening to James after he's been in the air for a long time.

e) Now that you have a velocity function v(t), integrate it with respect to t to find a position function h(t). This will introduce a second constant C₂.

f) Suppose James begins at a height of 3000m with no initial velocity, what is his height 3s later?

Answer 1