Chapter 9, Problem 1TP

Solution

a.

To determine the area of the triangle formed by the sun and the starting and ending positions of mars.

The area is $817.6045\text{ million square miles}\text{.}$

Given:

The planet’s position at different times is shown below:

  

Calculation:

Now, calculating the height of the triangle as:

  $\begin{array}{l}\sin \theta =\frac{h}{1.383}\\ h=1.383\sin \left(5.71°\right)\\ h=1.383\times 0.099\\ h=0.1369AU\end{array}$

Here,

  $\begin{array}{l}1AU=93\text{ million miles}\\ h=0.1369\times 93\\ h=12.732\text{ million miles}\end{array}$

Now, area of the triangle is calculating as:

  $\begin{array}{l}\text{Area}=\frac{1}{2}\times \text{base}\times \text{height}\\ A=\frac{1}{2}\times \left(1.38\times 93\right)\times 12.732\\ A=\frac{1635.209}{2}\\ A=817.6045\text{ million square miles}\text{.}\end{array}$

Hence, the area is $817.6045$ million square miles.

b.

To explain that as a planet moves closer to the sun along its orbit, what does Kepler’s law imply about the planet’s speed.

Interpretation:

Kepler’s law says it traverse equal areas in equal times. When the planet moves closer to sun, the planet’s speed must increase so as to cover the same area when the planet was traversing when it was far.