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Chapter 9: Problem 99

(See The Wide World of Fluids article "At \(12,600 \mathrm{mpg}\) It Doesn't cost Much to 'Fill 'er Up," section 9.3.3.) (a) Determine the power it takes to overcome aerodynamic drag on a small \(\left(6 \mathrm{ft}^{2}\right.\) cross section), streamlined \(\left(C_{D}=0.12\right)\) vehicle traveling \(15 \mathrm{mph}\) (b) Compare the power calculated in part (a) with that for a large \(\left(36 \mathrm{ft}^{2} \text { cross-sectional area }\right),\) nonstreamlined \(\left(C_{D}=0.48\right) \mathrm{SUV}\) traveling 65 mph on the interstate.

Short Answer

Expert verified
In this exercise, the most important part is carrying out conversions correctly and using the correct formulas. In conclusion, the power required to overcome aerodynamic drag is smaller for the streamlined vehicle than for the non-streamlined SUV.

Step by step solution

01

Compute drag force for the small vehicle

The drag force can be calculated using the formula \(D=0.5 \cdot C_D \cdot A \cdot p \cdot V^2 \). Given are \(C_D=0.12\), \(A=6 ft^2 \) (which we need to convert to \(m^2\) by multiplying it by \(0.092903\)), \(p = 1.184 kg/m^3\) which is the air density at sea level (20 degrees Celsius), and \(V=15 mph\) which also should be converted to m/s (multiply it by \(0.44704\)). So, \(D=0.5 \cdot 0.12 \cdot 6 \cdot 0.092903 \cdot 1.184 \cdot (15 \cdot 0.44704)^2 \).
02

Compute power for the small vehicle

To compute the power, the formula is \(P=D \cdot V\). Simply substitute the previously calculated drag force \(D\) and the given velocity \(V=15 mph\) which should be converted to m/s into the formula to get the power.
03

Compute drag force for the large SUV

Repeat step 1 but use the following parameters instead: \(C_D=0.48\), \(A=36 ft^2\), \(p = 1.184 kg/m^3\), \(V=65 mph\). Convert area and velocity to SI units then plug them into the formula.
04

Compute power for the large SUV

Like in step 2, calculate the power by substituting the calculated drag force from step 3 and the given velocity \(V=65 mph\) (converted to m/s) into the formula.
05

Compare powers

Now that the powers required for both vehicles have been calculated, they can be compared.

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