Question

Think \& Calculate You drive in a straight line at \(20.0 \mathrm{~m} / \mathrm{s}\) for \(10.0 \mathrm{mi}\), then at \(30.0 \mathrm{~m} / \mathrm{s}\) for another \(10.0 \mathrm{mi}\). (a) Is your average speed \(25.0 \mathrm{~m} / \mathrm{s}\), more than \(25.0 \mathrm{~m} / \mathrm{s}\), or less than \(25.0 \mathrm{~m} / \mathrm{s}\) ? Explain. (b) Verify your answer to part (a) by calculating the average speed.

Short answer

The average speed is less than 25.0 m/s.

Step-by-step solution

Convert Distance from Miles to Meters

The distances given are in miles, so we need to convert them to meters first. We know that 1 mile is equal to 1609.34 meters. Therefore, we multiply 10 miles by 1609.34 meters per mile, which gives us 16093.4 meters for each distance segment.

Calculate Time for Each Segment

For the first segment, where the speed is 20.0 m/s, use the formula \( t = \frac{d}{v} \) to find the time: \( t_1 = \frac{16093.4}{20} = 804.67 \) seconds. For the second segment at 30.0 m/s, \( t_2 = \frac{16093.4}{30} = 536.45 \) seconds.

Calculate Total Time

The total time is the sum of the time taken for each segment: \( t_{\text{total}} = t_1 + t_2 = 804.67 + 536.45 = 1341.12 \) seconds.

Calculate Average Speed

The average speed is calculated using the formula \( v_{\text{avg}} = \frac{d_{\text{total}}}{t_{\text{total}}} \), where \( d_{\text{total}} = 16093.4 + 16093.4 = 32186.8 \) meters. Thus, \( v_{\text{avg}} = \frac{32186.8}{1341.12} = 23.99 \) m/s.

Conclusion

Since the calculated average speed is 23.99 m/s, which is less than 25.0 m/s, the average speed is indeed less than 25.0 m/s.

Key concepts

Speed Calculation

Speed calculation is a fundamental concept in physics that helps us understand how fast an object is moving. To calculate the speed, you need to know two primary things: the distance traveled and the time it took to travel that distance.

The basic formula for speed is:
- Speed (\( v \)) = Distance (\( d \)) divided by Time (\( t \))

In mathematical terms, this can be expressed as:
\[v = \frac{d}{t}\]
This formula allows us to find out not just the immediate speed but can also be used to calculate the average speed over a journey. Average speed is particularly useful when an object's speed varies over time, as it gives a single value representing the whole trip. Remember, speed is a scalar quantity which means it has only magnitude and no direction.

This is important to grasp because in real-world scenarios, like driving or running, we rarely move at a constant speed. Calculating average speed can give us a clearer picture of the overall performance and efficiency of a journey.

Unit Conversion

Unit conversion is crucial when working with scientific calculations, especially in physics. Sometimes, the measurements you have may not match the units required for computations, so you need to convert them to a common unit to analyze the data correctly.

In the exercise, distances were given in miles, but speeds were in meters per second. To perform accurate calculations, converting miles to meters was necessary. Remember,

• 1 mile is equal to 1609.34 meters.

Having this standard conversion in mind helps when you encounter measurements with different unit systems.

When converting units, it's easy to make mistakes, so always double-check your conversions to avoid errors in your final calculations. Being adept at unit conversions streamlines the physics problem-solving process and ensures results are accurate and consistent.

Distance and Time

Distance and time are core components of motion and speed problems. They determine how objects move through space and over time, respectively.

Distance is the total length of the path traveled by the object, while time is the duration taken to cover that distance. In calculations, it's crucial to have consistent units when working with both distance and time, especially in speed calculations as seen in the exercise.

For example, the calculation in the exercise used the formula for time: \[ t = \frac{d}{v} \]where \( t \) is the time, \( d \) is the distance, and \( v \) is the speed. This formula helps determine the time required for each segment of the journey.

By adding the times together, you obtain the total time, which allows for computing the average speed later. Understanding the interplay between distance and time lets you accurately plan and predict journey results, making these concepts key in physics problem-solving.

Physics Problem Solving

In physics problem solving, following a structured approach is critical. It helps break down complex problems into manageable sections.

Here’s a basic roadmap for solving physics problems:

• Identify the problem and list knowns and unknowns.
• Select the relevant equations to use.
• Keep consistent units across all measurements.
• Perform calculations step by step, showing work clearly.
• Verify results with logical reasoning.

By using the step-by-step method laid out in the exercise, we could effectively solve the problem of calculating the average speed.

Each segment of the trip was tackled individually, calculating times first and summed up for total time, before finding the average speed. This structure helps avoid confusion, ensures precision, and enhances understanding of underlying physics concepts.