In their college years, Stevie G. was helping his roommate Biff G. move from Ankeny, IA, to Ames, IA. Steve and Biff put a queen-size Lebeda Heritage pillowtop mattress and box spring on the top of Biff’s 1999 Pontiac Grand Am sedan. To the disgust of Stevie G., Biff used baling twine to fasten the mattress and box spring combination to the roof of the car, which was centered on the roof. They had a choice of using US Highway 69 or US Interstate 35 in their travels. In this analysis, Stevie G. would like to have answers to the following questions: How far does the mattress hang over the roof of the car? What is the expected drag incurred by the mattress, and what is the lift exposed on the mattress for the two travel route options? What is the expected mileage decrease due to the mattress and box spring for the two travel options? How many times should the baling twine be wrapped around the roof of the car to ensure the mattress and box spring remained on the car, with a safety factor of 2.5?
The Correct Answer and Explanation is:
Answer and Explanation:
1. Mattress Overhang:
A queen-size mattress measures approximately 80 inches long and 60 inches wide. A 1999 Pontiac Grand Am sedan has a roof length of roughly 55 inches and a width of about 54 inches. If the mattress is centered on the roof, the overhang at the front and rear would be:80−552=12.5 inches\frac{80 – 55}{2} = 12.5 \text{ inches}280−55=12.5 inches
So, the mattress overhangs about 12.5 inches in front and back.
2. Drag and Lift Forces:
Drag force FdF_dFd is given by:Fd=12CdρAv2F_d = \frac{1}{2} C_d \rho A v^2Fd=21CdρAv2
Where:
- CdC_dCd (drag coefficient for a flat surface) ≈ 1.0
- ρ\rhoρ (air density) ≈ 1.225 kg/m³
- AAA (frontal area) ≈ 1.52 m² (based on mattress width and thickness)
- vvv is velocity in m/s
For US Highway 69 (speed ≈ 55 mph = 24.6 m/s):Fd≈0.5×1.0×1.225×1.52×(24.6)2≈563 NF_d \approx 0.5 \times 1.0 \times 1.225 \times 1.52 \times (24.6)^2 \approx 563 \, \text{N}Fd≈0.5×1.0×1.225×1.52×(24.6)2≈563N
For I-35 (speed ≈ 70 mph = 31.3 m/s):Fd≈0.5×1.225×1.52×(31.3)2≈912 NF_d \approx 0.5 \times 1.225 \times 1.52 \times (31.3)^2 \approx 912 \, \text{N}Fd≈0.5×1.225×1.52×(31.3)2≈912N
Lift force can be comparable in magnitude for flat surfaces, especially if the mattress catches wind beneath it. So lift ≈ drag, giving 563 N on Highway 69 and 912 N on I-35.
3. Mileage Decrease:
Increased drag reduces fuel efficiency. A typical penalty is:
- 10–15% loss at 55 mph
- 20–25% loss at 70 mph
So the expected mileage drop is:
- 10–15% on US-69
- 20–25% on I-35
4. Baling Twine Wraps with Safety Factor:
Assume combined weight of mattress and box spring = 150 lbs (≈ 667 N). With a safety factor of 2.5, force to resist = 1667.5 N.
If baling twine can safely hold 200 N per loop (depending on twine strength):1667.5200=8.34→round up to 9 loops\frac{1667.5}{200} = 8.34 \rightarrow \text{round up to } 9 \text{ loops}2001667.5=8.34→round up to 9 loops
Final Answer:
- Mattress overhangs by 12.5 inches front and back.
- Drag forces: 563 N on Hwy 69, 912 N on I-35
- Lift forces: roughly equal to drag.
- Mileage decrease: 10–15% on Hwy 69, 20–25% on I-35
- Minimum 9 wraps of baling twine needed for safety.
