General Astronomy, 29:61
Fall, 2004
Fifth Homework Set...September 27, 2004.
Due October 9, 2004

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 with a mass of $10^{22}$ kg is in orbit around the Sun. Its motion is defined in an $(x,y,z)$ coordinate system with the Sun at the origin. At a given moment, the vector $\vec{r}$ giving the position of the planet is

$$\vec{r} = 4.0 \hat{e}_x + 1.0 \hat{e}_y + 0 \hat{e}_z$$ (1)

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

$$\vec{v} = 2.0 \hat{e}_x + 5.0 \hat{e}_y + 0 \hat{e}_z$$ (2)

with units in kilometers/sec. Calculate the angle $\theta$ between the $\vec{r}$ and $\vec{v}$ vectors.

(2) For problem # 1, calculate the orbital angular momentum. Be sure and give the answer in proper SI units, and specify the direction as well as magnitude of the vector.

(3) Here's one similar to # 1, but with different $\vec{r}$ and $\vec{v}$, but the same mass. Now we have

$$\vec{r} = -2.0 \hat{e}_x + 3.0 \hat{e}_y + 0 \hat{e}_z$$ (3)

and

$$\vec{v} = -3.0 \hat{e}_x -1.0 \hat{e}_y + 0 \hat{e}_z$$ (4)

where again the distances are in astronomical units and the velocities in km/sec. Now what is the angle $\theta$ between the $\vec{r}$ and $\vec{v}$ vectors? Hint: Sketch the vectors in an $(x,y$ coordinate system, then proceed to answer the question.

(4) What is the angular momentum of # 3? Write it down as a vector. Does the angular momentum vector have the same direction as that in # 1 and # 2?

(5) Here is another problem with vector equations, like the one discussed in class on Monday, September 27. This one involves Newton's Second Law. An object with a mass of 2 kg is initially moving with velocity

$$\vec{v} = 5.0 \hat{e}_x + 2.0 \hat{e}_y + 0 \hat{e}_z \mbox{ m/sec }$$ (5)

. It is then acted upon by a force given by

$$\vec{F} = 0 \hat{e}_x + 0 \hat{e}_y + 1.0 \hat{e}_z \mbox{ Newtons }$$ (6)

After 5 seconds, what is the vector velocity?

(6) Here is one that you can use 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.

$$\vec{v} = -(a+bt) \hat{e}_x + (c-dt) \hat{e}_y + 0 \hat{e}_z$$ (7)

where $a,b,c,d$ are constants, and $t$ is the time. What is the vector acceleration?

(7) 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.

(8) 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.

(9) 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?