Course:PHYS350/Tutorial 3 2007W

0. You will derive the shape of orbits for a force

$\overrightarrow{F}=-\frac{GMm}{r^3}\overrightarrow{r}$

First, show that the following vector is constant for motion under gravity

$\overrightarrow{C}=\hat{r}-\frac{\overrightarrow{J}\times \overrightarrow{p}}{GM{m}^2}=\frac{\overrightarrow{r}}{\sqrt{\overrightarrow{r}\cdot \overrightarrow{r}}}-\frac{\overrightarrow{J}\times \overrightarrow{p}}{GM{m}^2}.$

You will probably find the following identity helpful:

$\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=\overrightarrow{b}(\overrightarrow{a}\cdot \overrightarrow{c})-\overrightarrow{c}(\overrightarrow{a}\cdot \overrightarrow{b}).$

Second, calculate $\overrightarrow{r}\cdot \overrightarrow{C}$ and calculate an equation in polar coordinates for the orbit (solve for the magnitude of $\overrightarrow{r}$ in terms of a function of $\theta$, the angle between $\overrightarrow{r}$ and $\overrightarrow{C}$). You will find the following result helpful:

$\overrightarrow{a}\cdot (\overrightarrow{b}\times \overrightarrow{c})=(\overrightarrow{a}\times \overrightarrow{b})\cdot \overrightarrow{c}.$

Third, what is the length of the semi-latus rectum, semimajor axis and the semiminor axis? What is the area of the ellipse? Using the conservation of angular momentum, calculate the period of the orbit. You should find that it depends on the semimajor axis alone.

Fourth, calculate $\overrightarrow{J}\cdot \overrightarrow{C}$ and $\overrightarrow{C}\cdot \overrightarrow{C}$ to find the value of the eccentricity in terms of the other constants of the motion. What is the length of the semimajor axis in terms of the energy of the orbit?

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The following problem will not count toward your grade for this tutorial. Rather, it will count as a half of a reading quiz, so if you attempt it you will be earning points against the final exam. There is nothing to lose.

24. A ballistic rocket (one that moves freely under gravity after its initial launch) is fired from the surface of the Earth with velocity $v<\sqrt{Rg}$ at an angle $\alpha$ to the vertical. (Ignore the Earth's rotation.) Find the equation of its orbit. Express the range $2R\theta$ (measured along the Earth's surface) in terms of the parameters $l$ and $a$, and hence show that to maximize the range, we should choose $\alpha$ so that $l=2a-R$. (Hint: A sketch might help.) Deduce that the maximum range is $2R\theta$ where

$\mathrm{s}\mathrm{i}\mathrm{n}\theta =\frac{v^2}{2Rg-v^2}.$

Given that the maximum range is 3600 nautical miles (one nautical mile is one arcminute along the Earth's surface), find the launch velocity and the angle at which the rocket should be launched.