A Reynolds number greater than about 1000 is considered high. A high Reynolds number means, as we’ve seen, that the drag arises primarily from inertia, with viscosity playing a minor role. In this case, drag occurs because the moving object has to push the fluid out of the way. As the example of the baseball suggests, the Reynolds number is high for most ordinary objects—balls, cars, planes—moving through air at ordinary speeds. Re is also high for fish and larger objects moving through water.

For high Reynolds numbers, the drag force for motion through a fluid at speed v is

Here $\rho$ is again the density of the fluid, A is the cross-section area of the object (in ${\text{m}}^2)$ as defined below, and the dimensionless drag coefficient $C_{\text{d}}$ depends on the object’s shape. More streamlined, aerodynamic shapes have lower values of $C_{\text{d}}$. TABLE 6.3 lists the drag coefficients for some common moving objects.

For a high Reynolds number, the magnitude of the drag force is proportional to the square of the object’s speed. If the speed doubles, drag increases by a factor of 4. This model of drag is often called quadratic drag.

The cross-section area A is the two-dimensional projection that you see when an object is coming toward you. If you were to shine a spotlight at the object as it approaches you, A is the area of the object’s shadow on a screen behind it. Many objects can be modeled as spheres or cylinders, so FIGURE 6.19 shows cross-section areas and drag coefficients for a smooth sphere and for two different orientations of a cylinder.

FIGURE 6.19 Cross-section areas and drag coefficients for a sphere and a cylinder.

EXAMPLE 6.7 Air resistance compared to rolling friction

A 1400 kg Toyota Prius, seen from the front, is 1.7 m wide and 1.5 m tall. Its drag coefficient, given in Table 6.3, is 0.24. At what speed does the magnitude of the drag equal the magnitude of the rolling friction?

MODEL Use the models of rolling friction and quadratic drag. Note that this is not a constant-force situation. Assume an air temperature of 20°C.

VISUALIZE FIGURE 6.20 shows the car and a free-body diagram. A full pictorial representation is not needed because we won’t be doing any kinematics calculations.

FIGURE 6.20 A car experiences both rolling friction and drag.

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SOLVE Drag is less than friction at low speeds, where air resistance is negligible. But drag increases as v increases, so there will be a speed at which the two forces are equal in size. Above this speed, drag is more important than rolling friction.

There’s no motion and no acceleration in the vertical direction, so we can see from the free-body diagram that $n=F_{\text{G}}=mg$. Thus $f_{\text{r}}={\mu }_{\text{r}}mg$. Equating friction and drag, we have

$${\displaystyle \frac{1}{2}}{C}_{\text{d}}\rho A{v}^2={\mu }_{\text{r}}mg$$

Solving for $v$, we find

$$v=\sqrt{{\displaystyle \frac{2{\mu }_{\text{r}}mg}{C_{\text{d}}\rho A}}}=\sqrt{{\displaystyle \frac{2\left(0.02\right)(1400\text{kg})\left(9.8\text{m}\text{/}{\text{s}}^2\right)}{\left(0.24\right)(1.2\text{kg}\text{/}{\text{m}}^3)(1.5\text{m}\times 1.7\text{m})}}}=27\text{m}\text{/}\text{s}$$

where the value of ${\mu }_{\text{r}}$ for rubber on concrete was taken from Table 6.1.

REVIEW $27\text{m}\text{/}\text{s}$ is approximately 60 mph, a reasonable result. This calculation shows that we can reasonably ignore air resistance for car speeds less than 30 or 40 mph. Calculations that neglect drag will be increasingly inaccurate as speeds go above about 40 mph.