Problem 58 - College Physics

Question

A Toyota Prius driving north at \(65 \mathrm{mi} / \mathrm{h}\) and a VW Passat driving south at \(42 \mathrm{mi} / \mathrm{h}\) are on the same road heading toward each other (but in different lanes). What is the velocity of each car relative to the other (a) when they are \(250 \mathrm{ft}\) apart, just before they meet, and (b) when they are \(525 \mathrm{ft}\) apart, after they have passed each other?

Step-by-Step Solution

Answer

The relative velocity is 156.93 ft/s in both scenarios.

Step 1: Convert Units

First, we need to convert velocities from miles per hour to feet per second.For the Toyota Prius:\[65 \text{ mi/h} \times \frac{5280 \text{ ft}}{1 \text{ mi}} \times \frac{1 \text{ h}}{3600 \text{ s}} = \frac{65 \times 5280}{3600} \text{ ft/s} \approx 95.33 \text{ ft/s}\]For the VW Passat:\[42 \text{ mi/h} \times \frac{5280 \text{ ft}}{1 \text{ mi}} \times \frac{1 \text{ h}}{3600 \text{ s}} = \frac{42 \times 5280}{3600} \text{ ft/s} \approx 61.6 \text{ ft/s}\]

Step 2: Apply Relative Velocity Concept

Relative velocity is calculated as the difference in velocities when objects move in opposite directions.The velocity of the VW Passat relative to the Toyota Prius is:\[V_{\text{relative}} = V_{\text{Prius}} + V_{\text{Passat}} = 95.33 + 61.6 = 156.93 \text{ ft/s}\]This results holds true regardless of the distance as it pertains to the concept of relative velocity.

Step 3: Conclusion for Both Scenarios

Since relative velocity calculation is not dependent on the distance between the vehicles, it remains the same whether they are 250 ft apart or 525 ft apart. Therefore, in both scenarios (just before they meet and after they pass each other), the relative velocity of each car with respect to the other is:\[156.93 \text{ ft/s}\]

Key Concepts

Unit Conversion

Whenever you're dealing with velocities, especially in physics problems, you might need to convert units to maintain consistency. In this case, the velocities of the cars given in miles per hour (mi/h) need to be converted to feet per second (ft/s). This allows for easier calculations and comparisons, especially when distances in these problems are often given in feet. Here's how the conversion works:

• First, understand that 1 mile is equivalent to 5280 feet.
• An hour (1 hr) consists of 3600 seconds.

Using these conversion factors, you can convert miles per hour to feet per second using the formula:\[V \, (\text{in ft/s}) = V \, (\text{in mi/h}) \times \frac{5280 \, \text{ft}}{1 \, \text{mi}} \times \frac{1 \, \text{h}}{3600 \, \text{s}}\]Plugging in the numbers for the Toyota Prius and VW Passat, we get:

• For the Toyota Prius: \[65 \times \frac{5280}{3600} \approx 95.33 \, \text{ft/s} \]
• For the VW Passat: \[42 \times \frac{5280}{3600} \approx 61.6 \, \text{ft/s} \]

Remembering and applying this conversion method can make it much easier to tackle similar problems in the future.

Velocity Calculation

Once the speeds of the objects are converted, the next step is to calculate individual velocities. This is important when dealing with related concepts such as relative velocity. The calculated velocities for the Toyota Prius and VW Passat were approximately 95.33 ft/s and 61.6 ft/s, respectively. Calculating these individual speeds allows us to further evaluate the next concept, which is the relative velocity of the two moving objects.

Opposite Directions

When two objects are moving towards each other, their velocities add up when calculating their relative velocity. This is a crucial concept in understanding relative motion. Since the Toyota Prius and VW Passat are driving in opposite directions (one heading north, the other south), you can determine the relative speed between them by summing their individual speeds. The relative velocity can be computed with:\[V_{\text{relative}} = V_{\text{Prius}} + V_{\text{Passat}} = 95.33 + 61.6 = 156.93 \, \text{ft/s}\]Here, despite traveling in different lanes, the key is their directional opposition which makes their velocities cumulative for relative calculation purposes. Opposite directions in movement illustrate how quickly one vehicle approaches the other from its own frame of reference.

Distance Independence

An intriguing aspect of relative velocity in this context is that it remains constant irrespective of the distance between the two objects. Whether the cars are 250 feet apart or 525 feet apart, the relative velocity calculated remains the same: 156.93 ft/s. Why is that? Distance independence results because relative velocity measures how fast one object is moving towards another and this measurement doesn't change unless the velocities themselves change (in other words, unless the cars speed up or slow down). Distances only impact notions of timing or pacing (how long it takes to cover the space), not the rate of closing speed between the two objects. Thus, in problems focusing on relative motion, once the speed calculations are complete, changes in spatial separation don't affect the result. This realization can simplify many dynamics problems by focusing solely on velocity changes.