
where
v_{T} = mg/b = the terminal velocity
γ = b/m = the damping rate.
where
F(t_{F}) = (v_{0y} + v_{T})(1  e^{γtF}).
Sketch graphs for these two functions as a function of t_{F}.
(b) The solution is at the intersection point of the two graphs. Since f is linear and the Taylor series for F shows that it starts of linearly and then slows its rate of increase, there is a possibility that the graphs will never cross (except at the origin). Let's consider for what values of the parameters there will be a nonzero solution. Assume that the angle is between 0 and 90^{o} and that all of the constants are positive. First discuss from a physical point of view and then demonstrate your result mathematically.
(c) Expand F to second order in γt_{F} ( = t_{F}/t_{b}, therefore a dimensionless number) and solve for t_{F}.
(d) Show that if b = 0 you get the known result for time of flight, t_{F} = 2v_{0y}/g.
(e) Notice that if you put back the original constants (b, m, g) your solution is not second order in b. Arrange your constants so that b is grouped with other constants to make a dimensionless number and find the correction to t_{F} to first order in this dimensionless number.
Last revision 12. October, 2004.