Question 1.3.4: The drag force on an object moving through a liquid or a gas......
The drag force on an object moving through a liquid or a gas is a function of the velocity. A commonly used model of the drag force D on an object is
D = \frac{1}{2} ρ A C_{D} v² (1)
where ρ is the mass density of the fluid, A is the object’s cross-sectional area normal to the relative flow, v is the object’s velocity relative to the fluid, and C_{D} is the drag coefficient, which is usually determined from wind-tunnel or water-channel tests on models.Curve A in Figure 1.3.6 is a plot of this equation for an Aerobee rocket 1.25 ft in diameter, with C_{D} = 0.4, moving through the lower atmosphere where ρ =0.0023 slug/ft³, for which equation (1) becomes
D = 0.00056v² (2)
a. Obtain a linear approximation to this drag function valid near v =600 ft/sec.
b. Obtain a linear approximation that gives a conservative (high) estimate of the drag force as a function of the velocity over the range 0 ≤ v ≤ 1000 ft/sec.
a. The Taylor series approximation of equation (2) near v =600 is
D = D|_{v=600} +\frac{dD}{d v}|_{v=600} (v−600) = 201.6 + 0.672(v−600)This straight line is labeled B in Figure 1.3.6. Note that it predicts that the drag force will be negative when the velocity is less than 300 ft/sec, a result that is obviously incorrect. This illustrates how we must be careful when using linear approximations.
b. The linear model that gives a conservative estimate of the drag force (that is, an estimate that is never less than the actual drag force) is the straight-line model that passes through the origin and the point at v =1000. This is the equation D =0.56v, shown by the straight line C in Figure 1.3.6.