James has just jumped out of an airplane. After he opens his parachute he experiences 2...

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?

Expert Solution

genius_generous answered 2 years ago

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