Newton's Second Law of motion can be written as m vd/dot = F (t, v) where Ff(t, v) is the sum of forces that act on the object and may be a function of the time t and the velocity
of the object, v. For our situation, we will have two forces acting on the object gravity,
F_G = acting in the downward direction and hence will be positive, and air resistance, F_A = – yn acting in the upward direction and hence will be negative. Putting all of this together into Newton's Second Law gives the following.
m vd/dot = mg-yn
To simplify the differential equation let's divide out the mass, m
. vd/dot = g – yn/m
This is a first order linear differential equation. The solution will give the velocity, v (in m/s), of a falling object of mass m that has both gravity and air resistance acting on it.
Let's assume that we have a mass of 3 kg and that y = 0.6. Plugging this into (1) gives the differential equation:
vd/dot = 9.8-0.2v
1 Find and classify the critical point(s) for this equation.
2Use Maple or sketch direction field for this problem
3 Use Maple or sketch the solution if initial velocity is 25 meters per s
second. Describe how the
behavior of velocity change? (Increasing/decreasing, what is v(t) approaching as time t approach infinity?)
4 For the initial velocity 58 m/s, describe the dynamics of velocity?
