Problem 21 - Physics for Scientists and Engineers with Modern Physics

Question

A light-year is the distance light travels in one year (at speed \(\left.=2.998 \times 10^{8} \mathrm{~m} / \mathrm{s}\right)\). (a) How many meters are there in 1.00 light-year? (b) An astronomical unit (AU) is the average distance from the Sun to Earth, \(1.50 \times 10^{8} \mathrm{~km} .\) How many AU are there in 1.00 light-year? (c) What is the speed of light in \(\mathrm{AU} / \mathrm{h} ?\)

Step-by-Step Solution

Answer

(a) \(9.461 \times 10^{15} \text{ meters}\); (b) \(63,070 \text{ AU}\); (c) \(7.195 \text{ AU/hr}\).

Step 1: Calculate Seconds in One Year

To find the number of meters in 1 light-year, we first need to determine the number of seconds in a year. There are 365 days in a common year, with 24 hours in a day, 60 minutes in an hour, and 60 seconds in a minute. Thus, the total number of seconds is calculated by the formula: \[\text{Seconds in a year} = 365 \times 24 \times 60 \times 60 = 31,536,000 \text{ seconds}\]

Step 2: Calculate Meters in One Light-Year

Next, we use the speed of light to find out how many meters light travels in 31,536,000 seconds. The formula for distance is speed multiplied by time: \[\text{Distance} = 2.998 \times 10^8 \, \text{m/s} \times 31,536,000 \, \text{s} \approx 9.461 \times 10^{15} \, \text{m} \] Therefore, 1 light-year is approximately \(9.461 \times 10^{15} \text{ meters}\).

Step 3: Convert Light-Year to Astronomical Units

An AU is the average distance from the Sun to Earth, given as \(1.50 \times 10^8 \text{ kilometers}\). First, convert this to meters: \[1.50 \times 10^8 \, \text{km} = 1.50 \times 10^{11} \, \text{m} \] Now, divide the total distance of 1 light-year in meters by the distance of 1 AU to find in how many AU this distance fits: \[\text{AU in light-year} = \frac{9.461 \times 10^{15} \, \text{m}}{1.50 \times 10^{11} \, \text{m/AU}} \approx 63,070 \text{ AU} \]

Step 4: Calculate Speed of Light in AU per Hour

First, note the speed of light in meters per second is \(2.998 \times 10^8 \text{ m/s}\). To express this in AU per hour, carry out the following conversion: 1. Convert meters per second to meters per hour by multiplying by the number of seconds in an hour (3600 s): \[2.998 \times 10^8 \, \text{m/s} \times 3600 \, \text{s/hr} = 1.07928 \times 10^{12} \, \text{m/hr}\]2. Convert meters per hour to AU per hour by dividing by the number of meters in an AU:\[\frac{1.07928 \times 10^{12} \, \text{m/hr}}{1.50 \times 10^{11} \, \text{m/AU}} \approx 7.195 \text{ AU/hr} \]

Key Concepts

Speed of Light

The speed of light is an essential constant in physics, representing how fast light travels through a vacuum. This speed is incredibly fast, precisely

• \(2.998 \times 10^8 \text{ meters per second (m/s)}\)

To better understand this, think about light originating from the Sun and reaching Earth in just about 8 minutes. Light, traveling at this extraordinary speed, allows us to see celestial objects as they were many years ago, thanks to the finite speed at which light travels.
To express the speed of light in different units, such as kilometers per second, simply convert meters to kilometers by recognizing that 1 kilometer = 1,000 meters. But remember, understanding the speed of light’s magnitude is crucial in fields ranging from astrophysics to global communications.

Astronomical Unit

An astronomical unit (AU) is a unit of length, often used to measure distances within our solar system. It is particularly convenient when expressing the average distance from the Earth to the Sun. This distance is approximately

• \(1.50 \times 10^8 \text{ kilometers} \)

This choice of measurement reflects the scale of objects within our solar system, as using regular distances such as kilometers can result in cumbersome, large numbers. By using the AU, comparisons and calculations can be performed more easily, keeping values manageable. When considering distances in light-years versus AUs, it's noted that 1 light-year is about 63,070 AUs, demonstrating the vastness of space beyond our solar system.

Distance Conversion

Converting between different units of distance—the light-year and the astronomical unit—is a critical aspect when dealing with astronomical distances. A light-year is the distance light covers in one year, calculated using the speed of light multiplied by the number of seconds in a year:

• About \(9.461 \times 10^{15} \text{ meters}\) per light-year

To put this in perspective, converting light-year distances into astronomical units helps astronomers express these immense distances within the solar system reference frame. The conversion is straightforward:

• 1 AU is \(1.50 \times 10^{11} \text{ meters}\)
• Therefore, 1 light-year is about 63,070 AUs

Moreover, converting the speed of light into different measurement units like AU/hr simplifies the understanding of light travel timing across astronomical distances without dealing with enormous numbers. This illuminates various aspects of comprehending the universe's vastness.