Problem 98 - Chemistry Principles and Practice

Question

The speed of light is \(3.00 \times 10^{8} \mathrm{~m} / \mathrm{s}\). Assuming that the distance from the Earth to the Sun is 93,000,000 miles, (a) how many light-years is this (see question 1.71 )? (b) How many minutes does it take for light to reach the Earth from the Sun?

Step-by-Step Solution

Answer

(a) 1.58 x 10^{-5} light-years; (b) 8.31 minutes.

Step 1: Convert Miles to Meters

First, we need to convert the distance from the Earth to the Sun from miles to meters. We use the conversion factor: \(1 \text{ mile} = 1609.34 \text{ meters}\). So, \(93,000,000 \text{ miles} = 93,000,000 \times 1609.34 \text{ meters}\).

Step 2: Calculate the Distance in Meters

Multiply 93,000,000 by 1609.34 to get the distance in meters: \(93,000,000 \times 1609.34 = 1.496 \times 10^{11} \text{ meters}\).

Step 3: Calculate Distance in Light-Years

One light-year is the distance light travels in a year. Calculate it using: \(1 \text{ year} = 365.25 \times 24 \times 3600 \text{ seconds}\). Then, \(1 \text{ light-year} = 3.00 \times 10^8 \times 31,557,600 \text{ seconds}\). Divide the distance from the Earth to the Sun by this value to find light-years.

Step 4: Compute Light-Year Conversion

\[1 \text{ light-year} = 9.461 \times 10^{15} \text{ meters}\]. The distance from the Earth to the Sun is \(\frac{1.496 \times 10^{11} \text{ meters}}{9.461 \times 10^{15} \text{ meters/light-year}} = 1.58 \times 10^{-5} \text{ light-years}\).

Step 5: Calculate Time for Light to Reach Earth

Time is calculated with the formula: \(\text{Time} = \frac{\text{Distance}}{\text{Speed}}\). Use the distance in meters and speed in meters per second. So, \(\frac{1.496 \times 10^{11} \text{ meters}}{3.00 \times 10^8 \text{ m/s}}\).

Step 6: Convert Time to Minutes

Compute \(\frac{1.496 \times 10^{11}}{3.00 \times 10^8} = 498.67 \text{ seconds}\). Convert seconds to minutes by dividing by 60, resulting in \(\frac{498.67}{60} \approx 8.31 \text{ minutes}\).

Key Concepts

Light speed calculations

The speed at which light travels is fundamental in physics calculations. Light speed, denoted as \(3.00 \times 10^{8} \text{ meters per second}\), is one of the most significant constants in physics. It enables scientists to predict how long it will take for light to travel across various distances. For example, knowing the speed of light allows us to compute the time it takes for light to travel from the Sun to the Earth, which is key for understanding astronomical distances.
Light speed calculations are crucial when solving problems related to relativity and cosmology. When dealing with these calculations, it is always important to ensure that the units used remain consistent, often requiring conversions for accuracy.

Distance conversion

Distance conversion is essential in physics to ensure that calculations are correct and understandable. Most international physics computations prefer using meters as the standard unit of distance. To convert miles to meters, you use the conversion factor: \(1 \text{ mile} = 1609.34 \text{ meters}\).

• This conversion factor lets you translate the distance from the Earth to the Sun: \(93,000,000 \text{ miles} \times 1609.34 = 1.496 \times 10^{11} \text{ meters}\).
• Being consistent with units allows for precise calculations and minimizes errors.

Understanding and performing these conversions accurately is a fundamental skill in scientific computations.

Time calculation

Time calculation is fundamental when determining how long light takes to travel a certain distance. In this scenario, we are interested in the travel time from the Sun to the Earth. To find this, use the formula:
\[\text{Time} = \frac{\text{Distance}}{\text{Speed}}\]
With our distance of \(1.496 \times 10^{11} \text{ meters}\) and the speed of light being \(3.00 \times 10^{8} \text{ m/s}\), you calculate the time:
\[\text{Time} = \frac{1.496 \times 10^{11}}{3.00 \times 10^{8}} = 498.67 \text{ seconds}\]
To convert seconds to minutes, divide by 60: \(\frac{498.67}{60} \approx 8.31 \text{ minutes}\).
Mastering time calculation techniques is important for analyzing any phenomena that involve distances covered in space.

Light-year conversion

A light-year is a distance measurement that denotes how far light travels within one year. It simplifies the understanding of astronomical distances. To find how many light-years are in a given distance, you must first determine the amount of meters in one light-year:
\(1 \text{ light-year} = 3.00 \times 10^8 \text{ m/s} \times 31,557,600 \text{ seconds/year} = 9.461 \times 10^{15} \text{ meters}\).
Subsequently, to convert the distance from the Earth to the Sun into light-years:
Divide the distance in meters by meters per light-year:
\(\frac{1.496 \times 10^{11} \text{ meters}}{9.461 \times 10^{15} \text{ meters/light-year}} \approx 1.58 \times 10^{-5} \text{ light-years}\).
Using light-years as a measurement unit provides a more comprehensible way to discuss the vast distances in space.