Problem 37 - Introductory Chemistry Atoms First

Question

Light travels extremely rapidly \((c=3.00 \times\) \(\left.10^{8} \mathrm{~m} / \mathrm{s}\right) .\) Suppose you had to travel 30 miles to work every day. If you traveled at the speed of light for the entire trip, how long would it take you to get to work (in seconds)? \([1 \mathrm{mile}=1.61 \mathrm{~km}]\)

Step-by-Step Solution

Answer

If you traveled at the speed of light for the entire trip, it would take about \(1.61 \times 10^{-4} seconds\) to get to work.

Step 1: Convert miles to meters

To convert the distance from miles to meters, we first need to convert miles to kilometers using the given conversion factor, and then convert kilometers to meters using the standard conversion factor. 1 mile = 1.61 km, so 30 miles = 30 * 1.61 km 1 km = 1000 m, so 30 * 1.61 km = 30 * 1.61 * 1000 m

Step 2: Calculate the time taken to travel the distance

We know that the speed of light is given by \(c=3.00 \times 10^{8} m/s\). Now, we can use the formula for time to find out how long it would take to travel the 30 miles at the speed of light. Time = Distance / Speed where Distance = 30 * 1.61 * 1000 m and Speed = \(3.00 \times 10^{8} m/s\) Therefore, Time = \( \frac{30 \times 1.61 \times 1000}{3.00 \times 10^{8}} s\)

Step 3: Calculate the final answer

Now, we can perform the calculation to get the final answer: Time = \( \frac{30 \times 1.61 \times 1000}{3.00 \times 10^{8}} s\) = \( \frac{48300}{3.00 \times 10^{8}} s\) = \(1.61 \times 10^{-4}s\) So, if you traveled at the speed of light for the entire trip, it would take about \(1.61 \times 10^{-4} seconds\) to get to work.

Key Concepts

Distance Conversion

Understanding how to convert distances between different units is crucial in many fields, including physics and engineering. When converting from one unit to another, it's important to use the correct conversion factors. In this exercise, we are converting distance from miles to meters. To begin with, we need to convert miles to kilometers because we have a given conversion factor that bridges these two units.

• 1 mile is equivalent to 1.61 kilometers.
• Thus, 30 miles is equal to \(30 \times 1.61\) kilometers.

Next, we convert kilometers to meters because our final calculation requires the distance in meters:

• 1 kilometer equals 1000 meters.
• This turns \(30 \times 1.61\) kilometers into \(30 \times 1.61 \times 1000\) meters.

This step-by-step approach ensures accurate and reliable conversions which is essential in precise calculations like those required for physics problems.

Time Calculation

Calculating the time it takes for an event to occur can be done using the simple formula:\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}}\]In this problem, we want to find out how long it would take to travel 30 miles at the speed of light. We've already converted the 30 miles into meters, which is 48,300 meters.The speed of light is an impressive \(3.00 \times 10^8\) meters per second. Using our formula:\[ \text{Time} = \frac{48,300}{3.00 \times 10^8}\]After performing this division, you will find that it takes about \(1.61 \times 10^{-4}\) seconds to travel the 30-mile trip at the speed of light. This calculation illustrates how incredibly fast the speed of light is, as such a short time is unimaginable for human-scale speeds.

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a more manageable form. It's particularly useful in fields requiring calculations with extreme values, such as astrophysics and quantum physics.Scientific notation expresses a number as a product of two factors:

• A decimal part: any number from 1 up to 10 (but not including 10),
• an exponent of ten, which indicates how many places the decimal moves.

For example, the speed of light is stated as \(3.00 \times 10^8\) meters per second. Here, '3.00' is the decimal part, and \(10^8\) indicates that you multiply '3.00' by 10 raised to the power of 8.In the solution, after calculating time, we get \(1.61 \times 10^{-4}\) seconds. This means:

• '1.61' is the decimal part,
• '\(10^{-4}\)' tells us the decimal moves four places to the left.

This notation simplifies handling significant figures and facilitates easier multiplication and division of large or small numbers.