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Chapter 3: Problem 66

Two cars are driving toward each other on a straight and level road in Alaska. The BMW is traveling at \(100.0 \mathrm{km} / \mathrm{h}\) north and the VW is traveling at \(42 \mathrm{km} / \mathrm{h}\) south, both velocities measured relative to the road. At a certain instant, the distance between the cars is \(10.0 \mathrm{km} .\) Approximately how long will it take from that instant for the two cars to meet? [Hint: Consider a reference frame in which one of the cars is at rest. \(]\)

Short Answer

Expert verified
Answer: It will take approximately 4.23 minutes for the two cars to meet.

Step by step solution

01

Find the relative velocity

Now we need to find the relative velocity of the two cars. Since they are moving toward each other, their velocities will add up when we calculate the relative velocity. That is: \(V_{relative} = V_{BMW} + V_{VW} = 100\,\text{km/h} + 42\,\text{km/h} = 142\,\text{km/h}\) So, the relative velocity between the two cars is \(142\,\text{km/h}\).
02

Calculate the time it takes for the cars to meet

To find the time it takes for the cars to meet, we can use the formula: \(time = \frac{distance}{velocity}\) Here, the distance between the cars is given as \(10\,\text{km}\), and we found the relative velocity to be \(142\,\text{km/h}\): \(time = \frac{10\,\text{km}}{142\,\text{km/h}} \approx 0.0704\,\mathrm{h}\) To convert the time to minutes, multiply by 60: \(time = 0.0704\,\mathrm{h} \times 60\,\frac{\text{min}}{\text{h}} \approx 4.23\,\text{min}\) So, it will take approximately \(4.23\,\text{minutes}\) for the two cars to meet.

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