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Zorluk: Çok zorUnit Conversions in Motion Problems

A military convoy is traveling along a straight highway at a constant speed of 45 miles per hour45\text{ miles per hour}. A scout vehicle leaves the rear of the convoy and travels forward to reach the front of the convoy. The total length of the convoy is 1.5 miles1.5\text{ miles}, and the scout vehicle travels at a constant speed of 88 feet per second88\text{ feet per second}. How many minutes does it take for the scout vehicle to reach the front of the convoy? (1 mile=5,280 feet1\text{ mile} = 5,280\text{ feet})

  1. A
    0.1
  2. B
    1.5
  3. 6Cevap
  4. D
    10

Cevap

6
The correct answer is found by equalizing the speed units and analyzing relative motion. First, converting the scout's speed gives 88 ft/s×(3600/5280)=60 mph88\text{ ft/s} \times (3600 / 5280) = 60\text{ mph}. Because the convoy is also moving forward at 45 mph45\text{ mph}, the relative closing speed is 60 mph−45 mph=15 mph60\text{ mph} - 45\text{ mph} = 15\text{ mph}. The scout needs to cover the 1.5 mile1.5\text{ mile} length of the convoy, which takes 1.5 miles/15 mph=0.1 hours1.5\text{ miles} / 15\text{ mph} = 0.1\text{ hours}. Finally, converting this to minutes yields 0.1×60=6 minutes0.1 \times 60 = 6\text{ minutes}.

Adım Adım Çözüm

1
Convert the scout vehicle's speed from feet per second to miles per hour.
88 ft/s×3,600 s1 hr÷5,280 ft1 mile=60 mph88\text{ ft/s} \times \frac{3,600\text{ s}}{1\text{ hr}} \div \frac{5,280\text{ ft}}{1\text{ mile}} = 60\text{ mph}.
To establish a common unit of speed for both vehicles before calculating relative motion.
2
Calculate the relative speed of the scout vehicle compared to the convoy.
60 mph−45 mph=15 mph60\text{ mph} - 45\text{ mph} = 15\text{ mph}.
Because both vehicles are moving in the same direction, subtracting the convoy's speed from the scout's speed gives the actual rate at which the scout is catching up (closing speed).
3
Determine the time required to cover the distance of the convoy in hours.
Time = Distance / Rate = 1.5 miles/15 mph=0.1 hours1.5\text{ miles} / 15\text{ mph} = 0.1\text{ hours}.
Applying the time, rate, and distance formula using the length of the convoy and the relative closing speed.
4
Convert the time from hours to minutes.
0.1 hours×60 minutes/hour=6 minutes0.1\text{ hours} \times 60\text{ minutes/hour} = 6\text{ minutes}.
The question explicitly asks for the answer in minutes.

Anahtar Kavram

Relative Motion and Unit Conversions

Alternatif Yöntem

Instead of converting everything to miles and hours, you can convert everything to feet and seconds. Convoy speed = 45 mph×(5280/3600)=66 ft/s45\text{ mph} \times (5280 / 3600) = 66\text{ ft/s}. Relative speed = 88 ft/s−66 ft/s=22 ft/s88\text{ ft/s} - 66\text{ ft/s} = 22\text{ ft/s}. Distance = 1.5 miles×5280=7,920 feet1.5\text{ miles} \times 5280 = 7,920\text{ feet}. Time = 7,920 ft/22 ft/s=360 seconds7,920\text{ ft} / 22\text{ ft/s} = 360\text{ seconds}. Finally, 360 seconds/60 seconds/minute=6 minutes360\text{ seconds} / 60\text{ seconds/minute} = 6\text{ minutes}.
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Unit Conversions in Motion Problems

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