Chapter 1, Problem 1.98QE

(a)

Interpretation Introduction

Interpretation:

The speed of light is $\text{3}\text{.00}\text{×}{\text{10}}^{\text{8}}\text{m/s}$.  Assume that the distance from the Earth to the Sun is $\text{93,000,000}\text{miles}$.  The number of light years has to be calculated.

Explanation of Solution

Given that light travels at a rate of $\text{3}\text{.00}\text{×}{\text{10}}^{\text{8}}\text{m/s}$.

The conversion factor is used to convert miters into miles.

  $\begin{array}{l}\text{1}\text{km}\text{=}{\text{10}}^{\text{3}}\text{m}\\ \text{1}\text{mi}\text{=}\text{1}\text{.609}\text{km}\\ \\ \text{Distance}\text{=}\text{3}\text{.00}\text{×}{\text{10}}^{\text{8}}\frac{\text{m}}{\text{s}}\text{×}\left(\frac{\text{1}\text{km}}{{\text{10}}^{\text{3}}\text{m}}\right)\text{×}\left(\frac{\text{1}\text{mi}}{\text{1}\text{.609}\text{km}}\right)\\ \text{=}\text{1}\text{.86}\text{×}{\text{10}}^{\text{5}}\frac{\text{mi}}{\text{s}}\end{array}$

The conversion factor is used to convert seconds into light year.

  $\begin{array}{l}\text{1}\text{hr}\text{=}\text{3600}\text{s}\\ \text{24}\text{hr}\text{=}\text{1}\text{day}\\ \text{1}\text{year}\text{(light}\text{year)}\text{=}\text{365}\text{.25}\text{days}\\ \\ \text{Distance}\text{=}\text{1}\text{.86}\text{×}{\text{10}}^{\text{5}}\frac{\text{mi}}{\text{s}}\text{×}\left(\frac{\text{3600}\text{s}}{\text{1}\text{hr}}\right)\text{×}\left(\frac{\text{24}\text{hr}}{\text{1}\text{day}}\right)\text{×}\left(\frac{\text{365}\text{.25}\text{days}}{\text{1}\text{year}}\right)\\ \text{=}\text{5}\text{.87}\text{×}{\text{10}}^{\text{12}}\frac{\text{mi}}{\text{light}\text{year}}\end{array}$

The distance from the Earth to the Sun is $\text{93,000,000}\text{miles}$.

  $\begin{array}{l}\text{3}\text{.00}\text{×}{\text{10}}^{\text{8}}\text{m/s}\text{=}\text{5}\text{.87}\text{×}{\text{10}}^{\text{12}}\text{miles/light}\text{year}\\ \text{light}\text{years}\text{=}\text{93,000,000}\text{miles}\text{×}\frac{\text{1}\text{light}\text{year}}{\text{5}\text{.87}\text{×}{\text{10}}^{\text{12}}\text{miles}}\\ \text{=}\text{1}\text{.58}\text{×}{\text{10}}^{\text{-5}}\text{light}\text{years}\end{array}$

(b)

Interpretation Introduction

Interpretation:

The speed of light is $\text{3}\text{.00}\text{×}{\text{10}}^{\text{8}}\text{m/s}$.  Assume that the distance from the Earth to the Sun is $\text{93,000,000}\text{miles}$.  The number of minutes it takes for a light to reach the Earth from the Sun has to be calculated.

Explanation of Solution

Given that light travels at a rate of $\text{3}\text{.00}\text{×}{\text{10}}^{\text{8}}\text{m/s}$.

The conversion factor is used to convert miters into miles.

  $\begin{array}{l}\text{1}\text{km}\text{=}{\text{10}}^{\text{3}}\text{m}\\ \text{1}\text{mi}\text{=}\text{1}\text{.609}\text{km}\\ \\ \text{Distance}\text{=}\text{3}\text{.00}\text{×}{\text{10}}^{\text{8}}\frac{\text{m}}{\text{s}}\text{×}\left(\frac{\text{1}\text{km}}{{\text{10}}^{\text{3}}\text{m}}\right)\text{×}\left(\frac{\text{1}\text{mi}}{\text{1}\text{.609}\text{km}}\right)\\ \text{=}\text{1}\text{.86}\text{×}{\text{10}}^{\text{5}}\frac{\text{mi}}{\text{s}}\end{array}$

The conversion factor is used to convert seconds into minutes.

  $\begin{array}{l}\text{1}\text{min}\text{=}\text{60}\text{s}\\ \\ \text{Distance}\text{=}\text{1}\text{.86}\text{×}{\text{10}}^{\text{5}}\frac{\text{mi}}{\text{s}}\text{×}\frac{\text{60}\text{s}}{\text{1}\text{min}}\\ \text{=}\text{1}\text{.12}\text{×}{\text{10}}^{\text{7}}\text{mi/min}\end{array}$

The distance from the Earth to the Sun is $\text{93,000,000}\text{miles}$.  The minutes it takes to reach light from Earth to Sun is

  $\begin{array}{l}\text{minutes}\text{=}\text{93000000}\text{miles}\text{×}\frac{\text{1}\text{min}}{\text{1}\text{.12}\text{×}{\text{10}}^{\text{7}}\text{miles}}\\ \text{=}\text{8}\text{.30}\text{×}{\text{10}}^{\text{14}}\text{minutes}\end{array}$