Newton's 2nd law, drag

Problem:

An object of mass m is launched from a stationary helicopter towards the Earth with the speed v0.   It experiences a force of air resistance F = -kv, where k is a positive constant.   The positive direction of all vector quantities is downward.
(a)  Draw a free body diagram showing the forces acting on the object.
(b)  What is the terminal speed of the object?
(c)  Find the speed of the object as a function of time, v0, m, g and k.
(d)  What is the direction of the acceleration of the object?

Solution:

Problem:

A particle of mass m falls subject to the pull of gravity, with acceleration g, and to the force of air resistance.  The particle is dropped from height z = z0.  The initial velocity is zero.  The force of air resistance may be modeled as linearly dependent on the speed, so that the height of the projectile satisfies 

md2z/dt2 = -mg - bdz/dt.

(a)  Solve the equation of motion for the particle's height z(t).  Find the terminal velocity.
(b)  Take the b --> 0 limit and show that usual free-fall solution is obtained.

Solution:

Problem:

The forces acting on a sky-diver of mass m are the force of gravity and the drag force due to the air.  Assume the drag force is proportional to the square of the speed.  Find the diver's velocity as a function of time, and the diver's terminal velocity vf.  Assume vi = 0.

Hint:  ∫dx/(a2 – x'2) = (1/a) tanh-1(x/a)

Solution:

Problem:

A boat with mass m is slowed by a drag force F(v).  Its velocity decreases according to the formula v(t) = c2(t – tf)2 for t ≤ tf, where c is a constant and tf is the time a which it stops.  Find the force F(v) as function of v.

Solution: