Problem 88 - University Calculus: Early Transcendentals

Question

If you average \(30 \mathrm{mi} / \mathrm{h}\) on a \(150-\mathrm{mi}\) trip and then return over the same 150 mi at the rate of \(50 \mathrm{mi} / \mathrm{h}\), what is your average speed for the trip? Give reasons for your answer.

Step-by-Step Solution

Answer

The average speed for the trip is 37.5 mi/h.

Step 1: Calculate Time for First Half

To determine the time taken for the first half of the trip, use the formula \( \text{time} = \frac{\text{distance}}{\text{speed}} \). For the first 150 miles at 30 miles/hour: \[ \text{Time} = \frac{150}{30} = 5 \text{ hours} \]

Step 2: Calculate Time for Second Half

For the return trip of 150 miles at 50 miles/hour, use the formula again: \[ \text{Time} = \frac{150}{50} = 3 \text{ hours} \]

Step 3: Determine Total Distance and Time

The total distance covered during the round trip is \( 150 + 150 = 300 \text{ miles} \) and the total time taken is \( 5 + 3 = 8 \text{ hours} \).

Step 4: Calculate Average Speed

The average speed for the entire trip can be calculated using the formula: \( \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \). Substituting these values gives: \[ \text{Average Speed} = \frac{300}{8} = 37.5 \text{ mi/h} \]

Key Concepts

Understanding the Distance-Speed-Time Relationship

The distance-speed-time relationship is a fundamental concept in physics that helps us understand motion. It connects how far an object travels (distance), how fast it is moving (speed), and how long it takes to get there (time). The basic formula is:\[ \text{Distance} = \text{Speed} \times \text{Time} \]This simple equation can be rearranged to solve for any of the three variables:- If you want to find distance, multiply speed by time - To find time, divide distance by speed- To calculate speed, divide distance by timeGrasping this relationship is crucial because it allows you to solve many real-world problems. Whether you are planning a road trip or calculating the time it will take to reach a destination, this formula is your go-to tool. Always remember to keep your units consistent when using the formula, or else your answer might not make sense.

Calculating a Round Trip

Round trips involve traveling from one location to another and then returning back. In our problem, the journey goes 150 miles in one direction and then comes back the same 150 miles. This makes the total round trip distance 300 miles. However, calculating average speed on a round trip isn't as simple as averaging the speeds of going and returning. You must use the total distance and total time spent traveling. Here's why:

• The first half was covered at 30 mi/h, taking 5 hours.
• The return trip was at 50 mi/h, completed in 3 hours.

Adding these gives a total time of 8 hours. To find the average speed of the entire journey, divide the total distance by the total time:\[ \text{Average Speed} = \frac{300 \text{ miles}}{8 \text{ hours}} = 37.5 \text{ mi/h} \]This step shows that average speed calculations must account for time, not just the distance or velocities, in each leg of the trip.

Unit Conversion in Physics

Unit conversion is vital when working with formulas in physics to maintain consistency and prevent errors. In our example, the units are consistent, using miles (mi) for distance and hours (h) for time. Sometimes, you may need to convert units, like switching from miles per hour (mi/h) to kilometers per hour (km/h) or vice versa. Knowing the right conversion factors simplifies this: - 1 mile = 1.60934 kilometers - 1 kilometer = 0.621371 miles To convert - From miles per hour to kilometers per hour, multiply by 1.60934 - From kilometers per hour to miles per hour, multiply by 0.621371 Staying consistent with units improves calculation accuracy. It's also an essential skill for comparing data across different measurement systems, especially where international standards differ.