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Chapter 2: Problem 17

Taking Earth's orbit to be a circle of radius \(1.5 \times 10^{8} \mathrm{km},\) determine Earth's orbital speed in (a) meters per second and (b) miles per second.

Short Answer

Expert verified
a) The orbital speed of Earth around the Sun is approximately 29780 meters per second. b) The speed in miles per second is approximately 18.5 miles per second.

Step by step solution

01

Plug in the Values

First, plug in the known values. The radius \(r\) is given as \(1.5 \times 10^{11}\) meters (since 1 km = 1000 m) and \(T\) is one year. However, since the speed is needed in meters per second and miles per second, the time \(T\) has to be in seconds. So, convert the time \(T\) in seconds by multiplying the number of days in a year (365) by the number of hours in a day (24), the number of minutes in an hour (60) and the number of seconds in a minute (60). Now, you can plug in the given and converted values into the formula.
02

Calculate the Speed in Meters per Second

Next, calculate the speed using the formula \(v = \frac{2\pi r}{T}\) with \(T = 365 \times 24 \times 60 \times 60\) seconds. This will yield the orbital speed of the Earth in meters per second.
03

Convert the Speed from Meters per Second to Miles per Second

Lastly, convert the speed from meters per second to miles per second. Use the conversion factor that 1 mile = 1609.34 meters. So, divide the calculated speed in meters per second by 1609.34 to get the speed in miles per second.

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