Chapter 9, Problem 104CR

To determine

To find:

The polar equation of the path of the asteroid and also find the distance between the asteroid and earth.

Answer to Problem 104CR

The polar equation of the path of asteroid is $r=\frac{12000000}{1-\sin \theta }$ and the distance between the asteroid and Earth is approximately $r\approx 5393876.913\text{ miles}$ .

Explanation of Solution

Given information:

An asteroid is about 6,000,000 miles from the Earth at its closest approach.

The vertex of the asteroid is at $\theta =-\frac{\pi }{2}$

In terms of the distance between the Earth and asteroid then $\theta =-\frac{\pi }{3}$

Calculation:

The distance between the vertex and the pole is $\frac{p}{2}$ , so

$\begin{array}{l}\frac{p}{2}=6000000\\ p=12000000\end{array}$

Since the path is parabolic, so the eccentricity of the parabola is $e=1$

The directrix is horizontal and is below the pole since the vertex is below the pole. Use $e\text{ and }p$ to write the equation of the parabola

$\begin{array}{l}r=\frac{ep}{1-e\sin \theta }\\ r=\frac{12000000}{1-\sin \theta }\end{array}$

Put $\theta =-\frac{\pi }{3}$ in the equation to find the distance between the Earth and the asteroid

$\begin{array}{l}r=\frac{12000000}{1-\sin \theta }\\ r=\frac{12000000}{1-\sin \left(-\frac{\pi }{3}\right)}\\ r=\frac{12000000}{1-\left(-\frac{\sqrt{3}}{2}\right)}\\ r=\frac{12000000}{\frac{2+\sqrt{3}}{2}}\\ r\approx 5393876.913\text{ miles}\end{array}$

Therefore, the polar equation of the path of asteroid is $r=\frac{12000000}{1-\sin \theta }$ and the distance between the asteroid and Earth is approximately $r\approx 5393876.913\text{ miles}$ .