Problem 111 - Introductory Chemistry Atoms First

Question

An object travels \(80.0 \mathrm{~m} / \mathrm{s}\). How fast is it traveling in miles per hour? \([1 \mathrm{~m}=3.28 \mathrm{ft}, 1\) mile \(=5280 \mathrm{ft}]\)

Step-by-Step Solution

Answer

The object is traveling at approximately \(178.2 \mathrm{~mph}\).

Step 1: Convert meters per second to feet per second #

To convert the given speed from meters per second to feet per second, we will use the conversion factor 1 m = 3.28 ft. We will multiply the given speed (80.0 m/s) by this conversion factor to obtain the speed in feet per second: \( (\text{80.0 m/s}) \times (\frac{\text{3.28 ft}}{\text{1 m}}) \)

Step 2: Calculate the speed in feet per second #

Now, we will perform the multiplication and get the speed in feet per second: \( (\text{80.0 }) \times (\text{3.28}) = 262.4 \text{ ft/s} \)

Step 3: Convert feet per second to miles per hour #

To convert the speed from feet per second to miles per hour, we will use the conversion factors 1 mile = 5280 ft and 1 hour = 3600 seconds. We will multiply the speed in feet per second (262.4 ft/s) by these conversion factors to obtain the speed in miles per hour: \( (\text{262.4 ft/s}) \times (\frac{\text{1 mile}}{\text{5280 ft}}) \times (\frac{\text{3600 s}}{\text{1 hour}}) \)

Step 4: Calculate the speed in miles per hour #

Perform the multiplication and get the speed in miles per hour: \( (\text{262.4 }) \times (\frac{1}{\text{5280}}) \times (\text{3600}) \approx 178.2 \text{ mph} \) So, the object is traveling at approximately 178.2 miles per hour.

Key Concepts

Meters per second to feet per second

Understanding how to convert meters per second (m/s) to feet per second (ft/s) is a useful skill in physics and everyday problem-solving. The conversion is straightforward and utilizes a specific conversion factor:

• 1 meter (m) equals 3.28 feet (ft).

To perform the conversion, multiply the speed in meters per second by the conversion factor. So, to convert 80.0 m/s:\[80.0 \, \text{m/s} \times \frac{3.28 \, \text{ft}}{1 \, \text{m}} = 262.4 \, \text{ft/s}\]This indicates that an object traveling at 80 meters per second is also moving at 262.4 feet per second. This conversion step is foundational and can be applied to various physics problems that require different units.

Feet per second to miles per hour

Once you've converted meters per second to feet per second, converting this to miles per hour (mph) requires understanding the relationship between feet and miles, and seconds and hours. The key conversion factors are:

• 1 mile equals 5280 feet.
• 1 hour equals 3600 seconds.

To convert from feet per second to miles per hour, use these conversion factors like this:\[262.4 \, \text{ft/s} \times \frac{1 \, \text{mile}}{5280 \, \text{ft}} \times \frac{3600 \, \text{s}}{1 \, \text{hour}}\]Carrying out these calculations allows us to find that 262.4 ft/s converts to approximately 178.2 mph. This step by step understanding of conversions between different speed units can be very valuable in physics and engineering applications.

Conversion factors

Conversion factors are constants or ratios used to express a quantity in terms of another unit. This process simplifies a comparison or combination of measurements. Here’s how they work:

• A conversion factor represents an equivalence between two units (e.g., 1 m = 3.28 ft).
• The factor is used as a fraction, typically set to one when written because it represents the same quantity in different units, e.g., \(\frac{3.28 \, \text{ft}}{1 \, \text{m}} = 1\).

Applying these involves multiplying the initial measurement by the conversion factor to get to the desired unit. This method, when applied consistently, makes it easy to adapt measurements and the results of calculations to different unit systems. Using conversion factors is integral to accurately solving physics and engineering problems and ensuring coherence in cross-unit calculations.

Physics problem solving

Successfully solving physics problems often involves breaking down the problem into smaller, manageable parts. Unit conversion is a critical part of this process when different units are used. Here’s a step-by-step process:

• Identify the units: Begin by identifying the units involved in the problem and the units required for your solution.
• Use conversion factors: Apply relevant conversion factors to change from one unit to another, as demonstrated in the conversion of m/s to mph.
• Check consistency: Ensure units remain consistent throughout calculations to avoid errors.
• Solve systematically: Approach each conversion and calculation step systematically to avoid mistakes.

In the context of our original exercise, understanding each conversion step equips you with the skills needed for a wide array of physics-related challenges. Fully grasping each concept transforms daunting physics problems into a series of simpler tasks, leading to precise and accurate solutions.