Problem 102 - Chemistry Matter and Change

Question

Evaluate the following conversion. Will the answer be correct? Explain. rate \(=\frac{75 \mathrm{m}}{1 \mathrm{s}} \times \frac{60 \mathrm{s}}{1 \mathrm{min}} \times \frac{1 \mathrm{h}}{60 \mathrm{min}}\)

Step-by-Step Solution

Answer

The calculation was incorrect; the correct conversion should yield a larger number.

Step 1: Understanding Unit Conversion

The task is to convert the speed from meters per second \(\left( \frac{\text{m}}{\text{s}} \right)\) to meters per hour \(\left( \frac{\text{m}}{\text{h}} \right)\). To do this, we must first convert seconds to minutes and then minutes to hours.

Step 2: Conversion from Seconds to Minutes

We multiply \(\frac{75 \text{ m}}{1 \text{ s}}\) by \(\frac{60 \text{ s}}{1 \text{ min}}\). The seconds cancel out, leaving us with: \[ 75 \times 60 \frac{\text{ m}}{\text{ min}} \] which simplifies to \(4500 \frac{\text{ m}}{\text{ min}}\).

Step 3: Conversion from Minutes to Hours

Next, we convert from meters per minute to meters per hour by multiplying by \(\frac{1 \text{ h}}{60 \text{ min}}\). The minutes cancel out, giving: \[ 4500 \times \frac{1}{60} \frac{\text{ m}}{\text{ h}} \] which simplifies to \(75 \frac{\text{ m}}{\text{ h}}\).

Step 4: Evaluate the Final Result

The final result from the conversion is \(75 \frac{\text{ m}}{\text{ h}}\). This indicates that there was an error in calculation or reasoning, as intuitively converting meters per second to meters per hour should result in a larger number due to the increase in unit of time. Therefore, the calculation was incorrect.

Key Concepts

Understanding Meters per Second

When measuring speed, sometimes we use the unit "meters per second" (m/s). This unit tells us how many meters an object travels in one second. It's a straightforward but essential way to describe movement over short periods.
For instance, if you're running at 5 m/s, you cover 5 meters every second you're in motion. That's helpful for understanding speed in activities like jogging or sprinting, where things happen quickly.
It's important to grasp the concept that the denominator (seconds) is time, and the numerator (meters) is distance. Recognizing this helps make sure any conversions are accurate.

Converting to Meters per Hour

The unit "meters per hour" (m/h) may be less common, but it demonstrates how much distance is covered in an entire hour. Understanding m/h involves multiplying the m/s value by 3600 (since there are 3600 seconds in an hour).
If we have a speed of 75 m/s and convert it to m/h, the correct conversion would be:

• Multiply 75 by 3600, since there are 60 seconds in a minute and 60 minutes in an hour (60 x 60 = 3600).
• This gives us a result of 270,000 m/h.

This significantly larger number makes sense because an hour is much longer than a second, so the distance traveled should notably increase.

Avoiding Conversion Errors

Conversion errors happen when incorrect units or incorrect conversion factors are used. To avoid these mistakes, it's crucial to verify the unit relationships before calculating. Errors like assuming the same conversion factor holds for all transformations can lead to incorrect outcomes.
In the given example, multiplying by the wrong factors resulted in a decreased value, which defies logical expectations that longer durations cover more distances. Make sure to:

• Double-check each step of your conversion.
• Understand that converting smaller to larger units generally results in larger numbers.

Always perform a reasonableness check: Does the answer make sense physically?

The Tool: Dimensional Analysis

Dimensional analysis is a methodical way to tackle unit conversion problems. It uses units as a guide to ensure each step of your calculation maintains logical consistency.
The technique involves:

• Writing down all the conversion factors you know.
• Connecting those factors in a chain that cancels out unwanted units.
• Evaluating if the end result unit is what you intended.

In more complex problems, this method provides a framework that makes complex unit conversions manageable. Remember, it's a foundational skill for ensuring precision in scientific and everyday calculations.