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Chapter 1: Problem 73

The distance from Earth to the Moon is approximately \(240,000 \mathrm{mi.}\) (a) What is this distance in meters? (b) The peregrine falcon has been measured as traveling up to 350 \(\mathrm{km} /\) hr in a dive. If this falcon could fly to the Moon at this speed, how many seconds would it take? (c) The speed of light is \(3.00 \times 10^{8} \mathrm{m} / \mathrm{s} .\) How long does it take for light to travel from Earth to the Moon and back again? (\boldsymbol{d} ) ~ E a r t h ~ t r a v e l s ~ a r o u n d ~the Sun at an average speed of 29.783 \(\mathrm{km} / \mathrm{s} .\) Convert this speed to miles per hour.

Short Answer

Expert verified
The distance from Earth to the Moon in meters is approximately \(386,241,600 \mathrm{m}\). It would take the falcon about 3,971,060 seconds to travel to the Moon at its dive speed. It takes light approximately 2.575 seconds to travel from Earth to the Moon and back. The Earth's average speed around the Sun is approximately 66,611.6 miles per hour.

Step by step solution

01

Part (a): Converting miles to meters

To convert miles to meters, we can use the conversion factor: \(1 \mathrm{mile} = 1,609.344 \mathrm{m}\). Multiply the distance in miles by the conversion factor: \(240,000 \mathrm{mi} \times 1,609.344 \frac{\mathrm{m}}{\mathrm{mi}} = 386,241,600 \mathrm{m}\) The distance from Earth to the Moon in meters is approximately \(386,241,600 \mathrm{m}\).
02

Part (b): Time for falcon to travel to the Moon

First, convert the falcon's speed from km/hr to m/s: \(350 \frac{\mathrm{km}}{\mathrm{hr}} \times \frac{1,000 \mathrm{m}}{1 \mathrm{km}} \times \frac{1 \mathrm{hr}}{3,600 \mathrm{s}} = \frac{350,000}{3,600} \mathrm{m/s} = 97.222\bar{2} \frac{\mathrm{m}}{\mathrm{s}}\) Now we can compute the time it would take the falcon to travel to the Moon by dividing the distance by the speed: \(t = \frac{386,241,600 \mathrm{m}}{97.222\bar{2} \frac{\mathrm{m}}{\mathrm{s}}} = 3,971,060\mathrm{s}\) It would take the falcon about 3,971,060 seconds to travel to the Moon at its dive speed.
03

Part (c): Time for light to travel to Moon and back

The speed of light is given as \(3.00 \times 10^8 \frac{\mathrm{m}}{\mathrm{s}}\). First, calculate how long it takes for light to travel to the Moon: \(t_{1} = \frac{386,241,600 \mathrm{m}}{3.00 \times 10^8 \frac{\mathrm{m}}{\mathrm{s}}} = 1.2875 \mathrm{s}\) Since light needs to travel back to Earth, this time must be doubled: \(t_{\text{roundtrip}} = 2t_{1} = 2 \times 1.2875 \mathrm{s} = 2.575 \mathrm{s}\) It takes light approximately 2.575 seconds to travel from Earth to the Moon and back.
04

Part (d): Converting Earth's speed to mph

To convert Earth's average speed from km/s to mph, use these conversion factors: \(1\mathrm{mi} = 1.609344\mathrm{km}\) and \(1\mathrm{hr} = 3600\mathrm{s}\): \(29.783 \frac{\mathrm{ km}}{\mathrm{s}} \times \frac{1 \mathrm{mi}}{1.609344 \mathrm{km}} \times \frac{3600 \mathrm{s}}{1\mathrm{hr}} = 66,611.6\frac{\mathrm{mi}}{\mathrm{hr}}\) The Earth's average speed around the Sun is approximately 66,611.6 miles per hour.

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