Homework Assignment 4
Physics 55
Made available:
Due in class on:
Wednesday, September 21, 2005
Friday, September 30, 2005
Problem 1: Should the United States spend billions of dollars each
year for space exploration?
The United States is currently spending about 15 billion dollars a year on space exploration, which includes
unmanned exploration (developing and maintaining new telescopes, developing and maintaining robotic
spacecraft to visit planets and comets, developing and maintaining satellites to study Earth, and supporting
the many scientists who collect and make sense out of the data obtained) and a much more expensive effort
for manned exploration (the space shuttle fleet, the International Space Station, and the infrastructure of
mainly NASA buildings and people to support this effort). In January of 2004, as part of his campaign
speech, President Bush proposed to put people back on the Moon by the year 2018 at a total cost of about
100 billion dollars. The Moon mission is viewed by the Bush administration as an essential step toward
sending humans to Mars a decade or two after a Moon base has been established.
You have been randomly selected as a citizen of, or as a visitor to, the US to make a three-minute or less
personal statement before Congress to help Congress decide what to do about space exploration in the future.
In a concise essay of at least 200 but not more than 900 words, write down what you would say in front
of Congress, arguing for an increase, status quo, or decrease in the space exploration budget based on how
you feel about the value of astronomical knowledge in your life and to society. Given the august body of
politicians that you will be addressing and their busy schedules, you should write simply, carefully, and with
focus if you want to influence Congress. You should also cite the source of any information that you quote
in your essay.
Instead of handing in your essay with this homework assignment, please share your essay with the rest of the
class (who will act as Congress) by posting your essay on the Physics 55 Blackboard page. You do this by
going to the course Blackboard page, by clicking on the button “Communication”, by next clicking on the
button “Discussion Board”, by next clicking on the Forum title “Essays For or Against Space Exploration”,
and finally by clicking on the button “Add New Thread” which will pop up a window where you can submit
your essay.
Your essay must be posted by 1 pm on Friday, September 30, to be obtain credit as part of this assignment.
Problem 2: Kepler’s Laws
1. Following Figure 3.18 on page 72 of the text, use two thumbtacks and some loops of string (e.g., dental
floss) to draw on a piece of paper two different ellipses whose major axes have the same length and
such that the two ellipses have one focus in common. On your drawing, indicate which ellipse has
the smallest eccentricity and which one has the largest. You should hand in your drawing with this
assignment.
According to Kepler’s third law, a planet following your elliptical orbits, with the Sun at the common
focus, will have the same orbital period despite their different shapes.
2. A certain asteroid is 2 AU from the Sun at perihelion and 6 AU from the Sun at aphelion.
(a) Determine the length in AU of the semimajor axis.
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(b) Determine the sidereal period of the orbit.
3. You can use your knowledge of Kepler’s laws to identify an elliptical orbit that will convey a spacecraft
from the Earth to Mars, The insight is to have the Sun at one focus, the spacecraft’s perihelion at the
Earth, and the spacecraft’s aphelion at Mars. After being launched from Earth, the spacecraft would
then coast along this ellipse until it reaches Mars, at which point it could fire a rocket to enter orbit
around Mars or to land.
(a) What is the value of the semimajor axis of this ellipse in units of AU? (Hint: Draw a picture.
You can assume that the orbits of Earth and Mars around the Sun are circles to good accuracy.)
(b) Calculate how long in days a one-way trip from Earth would take along your elliptical orbit.
Problem 3: Two Astronauts
At a certain moment, two astronauts are motionless in space, far from any rocket, planet or star. Including
their space suits and gear, astronaut A has a mass of 400 kg while astronaut B has a mass of 300 kg and
their centers of mass are 2 m apart.
1. What are the weights (in newtons) respectively of astronauts A and B?
2. In what direction and how fast will astronaut A be moving after ten seconds has passed? Hint: Explain
why the acceleration a of Astronaut A can be assumed to be constant to high accuracy over a tensecond period, in which case you can use the formula v = at, which is the speed v of an object after
an amount of time t has passed, when the object undergoes a constant acceleration a and the object
initially has zero speed.
3. Astronaut A is tired of waiting so long for gravity to pull him to Astronaut B so he can borrow a hand
tool. Astronaut A has two 5 kg flashlights and is able to throw a flashlight with a maximum speed
of 20 m/s. By throwing one flashlight then the other in rapid succession in a direction opposite to
Astronaut B at this maximum speed
(a) determine Astronaut A’s final speed toward B,
(b) and estimate how long it will take for A to meet up with B. For this latter problem assume that
the astronauts are point particles of zero width that are initially 2 m apart.
Hint: Use the conservation law of momemtum: if m is the mass of a flashlight and M is the mass of
astronaut A, then 2m + M is the total mass of the two flashlights and of astronaut A before he throws
the first flashlight and the speeds of the flashlights and astronaut are both initially zero. After the
astronaut throws the first flashlight, the flashlight is moving away from the astronaut with a speed
of 20 m/s so you can get the speed of the astronaut plus second flashlight from this equation stating
momentum conservation:
0 = initial momentum of astronaut A carrying two flashlights
= −20m + v1 (m + M )
= momentum of one flashlight and momentum of “astronaut + flashlight”,
(1)
(2)
(3)
where v1 is the speed of the “flashlight + astronaut” which you can now solve for using this equation.
You then need to repeat this one more time for the astronaut throwing his second flashlight away to
get the final speed v2 of astronaut A.
This problem also illustrates how a rocket works, which is like an astronaut with a huge supply of
small flashlights who throws flashlights one after another to build up some speed. From this astronaut
problem, you can appreciate that neither the astronaut or rocket has to “push off” something to start
moving, it is all a matter of momentum conservation.
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Problem 4: Using Newton’s Version of Kepler’s Law to Determine
the Mass or Period of Some Object
Do problems 26b and 26c on page 150 of the text.
Problem 5: Optional Extra Credit Problems
1. Explain why, if you hold your breath and stand very still on an accurate scale, you will find that your
weight is not constant but oscillates around the value of your average weight. What do you predict for
the approximate period of this oscillation?
Also, explain whether your weight will increase or decrease just as you begin to step off the scale (and
test your prediction).
2. If the Earth were suddenly brought to a halt in its orbit around the Sun and then released, how long
would it take to fall into the Sun?
Hint: You can answer this problem without using calculus if you think creatively about Kepler’s laws.
A similar line of reasoning can answer the above question of how long it would take Astronaut A to
collide with Astronaut B under the influence of their mutual gravitational attraction.
Problem 6: Comments about the Homework and Course
• About how long did this assignment take to complete?
• Do you feel that you are understanding the course material? If not, please indicate what topics or
ideas you would like to understand better.
• Other comments or suggestions about the homeworks, lectures, or observation sessions?
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