Problems for Intermediate Methods in Theoretical Physics Edward F. Redish

Time of Flight: 2

In assignment 4, problem 2, we derived the equation for the time of flight, tF, of a projectile launched with a velocity v, launched at an angle θ from the vertical when viscous drag was included. The result was the equation

where

v0y = v0 cosθ = the y-component of the initial velocity

vT = mg/b = the terminal velocity

γ = b/m = the damping rate.

(a) Consider how you might solve this equation numerically (if you had values for the constants) by considering it as the intersection of two function graphs:

f(tF) = F(tF)

where

f(tF) = gtF

F(tF) = (v0y + vT)(1 - etF).

Sketch graphs for these two functions as a function of tF.

(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 non-zero solution. Assume that the angle is between 0 and 90o 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 γtF ( = tF/tb, therefore a dimensionless number) and solve for tF.

(d) Show that if b = 0 you get the known result for time of flight, tF = 2v0y/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 tF to first order in this dimensionless number.

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