Problem 14 - College Physics

Question

While riding on a bus traveling down the highway, you notice that it takes 2 min to travel from one roadside mile marker to the next one. (a) What is your speed in \(\mathrm{mi} / \mathrm{h} ?\) (b) How long does it take the bus to travel 100 yds?

Step-by-Step Solution

Answer

(a) 30 mi/h; (b) about 6.82 seconds.

Step 1: Convert time from minutes to hours

First, let's convert the time it takes to travel one mile from minutes to hours. We know it takes 2 minutes to travel one mile. To convert minutes to hours, use the conversion factor: 1 hour = 60 minutes. Thus, the time in hours is \(\frac{2}{60}\) hours.

Step 2: Calculate speed in miles per hour

Now that we have the time in hours, calculate the speed in miles per hour (mi/h). Speed is distance divided by time. Since the distance is 1 mile and the time is \(\frac{2}{60}\) hours, the speed is \(\frac{1}{\frac{2}{60}} = \frac{1 \times 60}{2} = 30\) mi/h.

Step 3: Convert yards to miles

For part (b), we need to find out how long it takes to travel 100 yards. First, convert yards to miles. There are 1760 yards in a mile. Thus, 100 yards is equal to \(\frac{100}{1760}\) miles.

Step 4: Calculate time taken to travel 100 yards

Using the speed from part (a), calculate the time taken to travel the 100 yards. Time is distance divided by speed. The speed is 30 mi/h and the distance is \(\frac{100}{1760}\) miles. Thus the time is \(\frac{\frac{100}{1760}}{30}\) hours.

Step 5: Convert time from hours to seconds

Convert the time calculated in Step 4 from hours to seconds because time in hours will be a small fraction. Use the conversion factor: 1 hour = 3600 seconds. So, \(\frac{100}{1760 \times 30} \times 3600 = \frac{100 \times 3600}{1760 \times 30}\) seconds. Calculate this value to find the time in seconds.

Key Concepts

Distance Conversion

Distance conversion is an essential technique used in physics to understand and translate distances into different units. Consider a situation where you need to convert yards into miles—a common requirement in physics problems. The factor between these two units is crucial for accuracy.

• There are 1760 yards in a mile.
• To convert yards into miles, divide the number of yards by 1760.

For instance, if you have 100 yards, you'll convert it by calculating \( \frac{100}{1760} \) miles. This conversion is vital when dealing with different types of units, as it allows precise comparison and calculation of distances. Whenever you see yards or other non-standard units in a problem, remember to convert them first to the units you are working with, such as miles in this case.

Time Conversion

Time conversion plays a pivotal role in simplifying physics problems. Frequently, it's necessary to switch between different time scales, such as minutes to hours or hours to seconds, for uniformity in calculation.

• Convert minutes to hours by dividing by 60.
• For instance, 2 minutes equals \( \frac{2}{60} \) hours or \( \frac{1}{30} \) hours.
• Convert hours to seconds by multiplying by 3600, since there are 3600 seconds in an hour.

For example, if you have a time in hours that is a small fraction, like \( \frac{100}{1760 \times 30} \) hours, multiply it by 3600 to get the equivalent time in seconds for easier interpretation. Whether solving a homework assignment or conducting an experiment, ensuring all time units are consistently measured makes calculations manageable and results comprehensible.

Physics Problem Solving

Solving physics problems often involves systematic steps to arrive at an answer. This methodology ensures clarity and accuracy in your results.

• Identify the knowns and unknowns: Clearly distinguish what information is provided and what you need to find.
• Use appropriate formulas: In our scenario, speed is calculated using \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \).
• Apply conversion factors: Make sure all measurements are in the correct units, such as converting first to determine a consistent time and distance unit.

Taking these actions step-by-step helps in breaking down complex problems. For instance, when calculating speed from minutes to miles per hour or determining the time taken to cover a certain distance, following a structured approach guarantees that each part of the problem is handled effectively, leading to the correct solution.

Unit Conversion in Physics

In physics, dealing with different units is commonplace and mastering unit conversion is key to solving many problems. This practice involves using conversion factors to switch quantities from one unit to another.

• Be aware of conversion factors: Know the basic conversions, like 1 mile = 1760 yards or 1 hour = 3600 seconds.
• Apply these factors carefully: Always align units with each part of your calculation to ensure consistency.

For example, converting miles per hour to a time fraction in seconds directly impacts accuracy. Translate all data into compatible units before inserting them into your equations. Practicing unit conversion not only helps you tackle physics exercises smoothly but also builds a foundational skill beneficial in various scientific inquiries.