General Astronomy, 29:61
Fall, 2005
Fifth Homework Set...September 23, 2005.
Due September 30, 2005

Show calculations and give reasons for your answers. Be sure to put the answers in the right physical units. Note: Data for some of these problems can be found in the appendix of the textbook.

(1) An object is in orbit around the Sun. Its motion is defined in an $(x,y)$ coordinate system with the Sun at the origin. At a given moment, the vector $\vec{r}$ giving the position of the planet is

$$\vec{r} = 4.0 \hat{e}_x + 1.0 \hat{e}_y$$ (1)

where the units are in astronomical units. The planet is moving with a velocity vector

$$\vec{v} = 2.0 \hat{e}_x + 5.0 \hat{e}_y$$ (2)

with units in kilometers/sec. Is the object moving in a circular orbit? Calculate the angle $\theta$ between the $\vec{r}$ and $\vec{v}$ vectors.

(2) Here is one that you can use your differential calculus on, although it can be solved with algebra as well. An object is moving with the following velocity as a function of time.

$$\vec{v} = -(a+bt) \hat{e}_x + (c-dt) \hat{e}_y + 0 \hat{e}_z$$ (3)

where $a,b,c,d$ are constants, and $t$ is the time. What is the vector acceleration? Hint: If a function is given by $f(t)=at^n$, the derivative of $f$ with respect to time is

$$\frac{df}{dt}=nat^{n-1}$$ (4)

(3) Compute the speed of an object in low Earth orbit, like the space shuttle. For a low Earth orbit, you may assume that the orbit is circular, with a radius equal to that of the Earth. Be sure and consult the Appendix for data which you will need.

(4) Look at Figure 3-1 of the book. What is the eccentricity of the orbit plotted there? For this problem, get yourself a plastic ruler and measure off the numbers you need.

(5) Let's continue with Figure 3-1. Assume that it shows the orbit of an object in the solar system and that the scale of the figure is 1cm = 1 astronomical unit. (a) What is the distance at perihelion (i.e. when the object is closest to the Sun)? (b) What is the distance at aphelion (i.e furthest from the Sun)? (c) What is the orbital period?

(6) Two masses, each of 10 kilograms, are 10 cm (10cm = 0.1m) apart. What is the gravitational force between them? I want a number! With the right units!.

(7) Keep thinking about the two masses in the previous problem. One of the masses is fixed (i.e. bolted to the wall) and the other is free to move. How does it move? Give a quantitative answer.

(8) Approximate the orbit of Jupiter by a circular orbit. (a) Using data from the book, calculate the speed at which Jupiter moves in its orbit. (b) Using your result from (a), calculate the orbital period. Compare your answer with the true answer.