Problem 103 - Introductory Chemistry Atoms First

Question

A train traveling at \(45.0\) miles \(/ \mathrm{h}\) has to make a trip of \(100.0\) miles. How many minutes will the trip take? Use unit analysis to calculate your answer, and show your work.

Step-by-Step Solution

Answer

The trip will take approximately \(133.33 \: \text{minutes}\).

Step 1: Write down the given values

Speed: \(45.0 \: \text{miles/h}\) Distance: \(100.0 \: \text{miles}\)

Step 2: Find the time in hours

We can calculate the time taken in hours by using the formula time = distance / speed. Time (hours) = \(\frac{100.0 \: \text{miles}}{45.0 \: \text{miles/h}}\)

Step 3: Calculate the time in hours

Time (hours) = \(\frac{100.0}{45.0}\) Time (hours) ≈ \(2.22 \: \text{hours}\)

Step 4: Convert the time from hours to minutes

We can convert the time from hours to minutes by multiplying the time in hours by 60 (since there are 60 minutes in an hour). Time (minutes) = \(2.22 \: \text{hours} \times 60 \frac{\text{minutes}}{\text{hour}}\)

Step 5: Calculate the total time in minutes

Time (minutes) = \(2.22 \times 60\) Time (minutes) ≈ \(133.33 \: \text{minutes}\) The trip will take approximately 133.33 minutes.

Key Concepts

Speed and Distance Relationship

Understanding the relationship between speed, distance, and time is fundamental in problems involving motion. Often expressed through the formula \( \, \text{Speed} = \frac{\text{Distance}}{\text{Time}} \, \), this relationship tells us how these three variables interact:

• **Speed** is the rate at which an object covers distance, typically measured in units like miles per hour (mph) or kilometers per hour (km/h).
• **Distance** represents how far an object travels, measured in units such as miles, kilometers, or meters.
• **Time** is the duration it takes to travel that distance, usually measured in hours, minutes, or seconds.

In our train exercise, you're given the speed and the distance, and you need to find the time the train will take for the journey. By rearranging the formula, we can solve for time: **Time = Distance ÷ Speed**.
This gives a clear linear relationship: increasing speed decreases travel time for a set distance, while a longer distance increases travel time at the same speed.

Unit Conversion

Unit conversions are essential when calculating different measurement units in problems like the train journey exercise. Converting units helps align dimensions to ensure calculations are straightforward and accurate.

• To convert time from hours to minutes, you multiply by 60, as there are 60 minutes in an hour.
• For conversions involving speed or distance, remember to keep units consistent—convert miles to kilometers if necessary, or vice versa.
• Maintaining consistent units helps avoid errors and make calculations more intuitive.

In our train problem, we convert the time calculated in hours to minutes for a more practical understanding of the journey's duration. Since the time is approximately \(2.22\) hours, multiplying by 60 gives \(2.22 \times 60 \approx 133.33\) minutes. This conversion ensures that the time aligns with common everyday understanding.

Time Calculation

Calculating time accurately is crucial in everyday scenarios, such as planning journeys or managing schedules. In the train exercise, after finding the time in hours using the formula \( \, \text{Time} = \frac{\text{Distance}}{\text{Speed}} \, \), the next step is converting that time into a more relatable unit.Steps to Calculate Time:

• Begin by determining the necessary units for the problem at hand — in this exercise, that was hours first.
• Use the given values to perform the calculation: here, \( \frac{100.0 \, \text{miles}}{45.0 \, \text{miles/h}} \) gives \( \approx 2.22 \, \text{hours}\).
• To convert hours into minutes, multiply the time by 60. This provides a final answer that's easily understandable.

This conversion into minutes makes the expected travel duration tangible and practically useful. Remember, time can often dictate much of our daily choices, from simple tasks to complex scheduling.