Chapter 6.1, Problem 105AYU

To determine

To find:the linear speed of the moon and express in miles perhour.

Answer to Problem 105AYU

The linear speed of the moon is $v\approx 2292\frac{\text{miles}}{\text{hour}}$

Explanation of Solution

Given information:

The given mean distance of the moon from earth (radius) is $\text{2}\text{.39}\times {\text{10}}^5\text{miles}$

The orbit of the moon around earth to takes $1$ revolution in $27.3$ days.

Concept used:

The object travels around the circles;

That an object moves around a circles of radius $r$ at a constant speed .If $s$ is the distance travelled in time $t$ around this circles, then the linear speed $v$ of the object its defined as.

  $v=\frac{s}{t}$

The relation between linear speed and angular speed;

  $v=r\omega$

Here $\omega$ is measured in radius per unit time.

The angular speed $\omega$ of this object is the angle $\theta$ (measures in radians) swept out. Divided by the elapsed time $t$ that is

  $\omega =\frac{\theta }{t}$

The radius is the mean distance of the moon from earth

  $r=2.39\times {10}^5\text{miles}$

The moon rotates around the earth to complete the one revolution in $27.3\text{days}$

  $1\text{revolution}=2\pi \text{radians}$

The central angle of the moon from the earth is

  $\theta =2\pi \text{radians}$

Convert the days in hours.

  $\begin{array}{l}t=27.3\text{days}\\ 27.3\text{days}=27.3\times 24\text{hours}\\ \left(1\text{day=24}\text{hours}\right)\\ t=655.2\text{hour}\end{array}$

The linear speed is.

  $v=\frac{s}{t}$

The distance travelled with radius is calculated as,

  $s=r\theta$

The radius is $r=2.39\times {10}^5\text{miles}$ and the angle is $\theta =2\pi \text{radians}$

The distance of the moon from earth is,

  $\begin{array}{l}s=2.39\times {10}^5\times 2\pi \\ s=4.78\pi \times {10}^5\text{miles}\end{array}$

The linear speed is calculated as,

  $\begin{array}{c}v=\frac{s}{t}\\ \text{=}\frac{4.78\pi \times {10}^5}{655.2}\frac{\text{miles}}{\text{hours}}\\ v=2291.94\frac{\text{miles}}{\text{hours}}\end{array}$

Therefore, the linear speed of the moon is $v\approx 2292\frac{\text{miles}}{\text{hours}}$