VPython tutorial 2 - University of Cape Town

Name:
Student Number:
University of Cape Town
Department of Physics
PHY1004W
VPython Tutorial
Voyage to the Moon
6 April 2010
Work in groups of 2. Carefully read and complete the following worksheet.
Show your worksheet to a demonstrator before you leave this afternoon. Satisfactory completion of this
task will result in 5 marks (!) being allocated to your laboratory year mark.
The Ranger 7 spacecraft was launched by NASA on 28 July 1964 with the purpose of transmitting high
resolution photographs of the surface of the moon while on a trajectory directly toward it. The
spacecraft transmitted more than 4 000 photographs of excellent quality during its last 17 minutes of
flight, before impacting the moon, on 31 July 1964.
In this tutorial, we will consider a very rough model of the Ranger 7 moon mission.
Part A. Sit at a lab bench … you will move to a PC later.
1. Say that the positions of the Earth and Moon are fixed in space, and there is a spacecraft in a stable
circular orbit around the Earth as illustrated below. The spacecraft is launched on a trajectory to the
moon by accelerating it to a velocity which allows it to escape its orbit around the Earth.
(a) The path of the spacecraft to the moon depends on its launch velocity.
Sketch on the diagram above a possible path that the spacecraft could follow, from its launch to its
impact on the surface on the moon. Label your sketch clearly.
(b) Describe the path in (a) in terms of the underlying physics concepts.
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2.
(a) The spaceship is shown at two positions on the diagram below. Illustrate the forces acting on the
spacecraft at positions A and B, using appropriate vector representations and labels.
A
B
(b) Clearly explain how the forces acting on the spacecraft change as it moves from position A to
position B.
(c) Write down the mathematical expressions of these forces.
3. Write down the mathematical expressions which describe the change in the position and momentum
of the spacecraft as it moves from the earth to the moon.
4. How long do you think the voyage will take? Explain your reasoning.
STOP !
… call a demonstrator who must check your answers and sign here ___________
before you move to a PC.
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Part B … now move to a PC
We will now write a VPython program to solve the equations of motion for a spacecraft
on a voyage from the Earth to the Moon.
I. Program organization
The program will have the same basic structure as in previous VPython projects:
1. An initialisation section in which: (a) numerical constants are assigned values; (b) 3D simulation
objects are created; and (c) the initial conditions — positions and momenta — are specified.
2. A loop — usually a while loop — in which: (a) the force on the objects is calculated; (b) the
momenta and positions are updated; and (c) the simulation and time is updated.
II. Getting started
II.1. Creating a new program file
You have already written a program for the motion of the Earth around the Sun; perhaps you
have extended it to the case of a planet and two suns. In either case it is a good starting point for
this program: read it in using IDLE and save it with a suitable name. You can then modify it rather
that rewriting from scratch.
Note: some of you are saving your files with strange names in odd places: save files under “My
Documents” or in a folder located in this folder. You must remember to include the “.py” extension
in the name, otherwise your code will not appear in colour. Also, choose a name that makes some
sense to you, and is meaningful, e.g. “moonvoyage.py”.
II.2. Creating the earth, moon and spacecraft
Some data: the mass of the Earth is M E = 6 × 10 24 kg, mass of the Moon is M M = 7 × 10 22 kg,
radius of the Earth is R E = 6.4 × 10 6 m and radius of the moon is R M = 1.75× 10 6 m. The distance
from the Earth to the Moon is 4.0 × 10 8 m. The gravitation constant is G = 6.67 × 10 −11 N m 2 kg -2 .
Take the mass of the spacecraft as 175 kg.
Modify your previous program to use the new data: it should start something like
from visual import *
## objects
earth = sphere( pos=(0, 0, 0), radius=6.4e6, color=color.blue )
moon = sphere( pos=(4.0e8, 0, 0), radius=1.75e6, color=color.white )
craft = sphere( pos=(...), radius=1.0e6, color=color.red )
##constants
G = 6.67e−11
earth.M = 6.0e24
moon.M = 7.0e22
craft.M = 175.0
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Create a “trail”, using the curve object, to represent the path of the spacecraft.
craft.trail = curve( pos=craft.pos, color=craft.color )
II.3 Initial values of variables
We will take the positions of the Earth and Moon to be fixed: the voyage will not take long — a
few days, and the motion of the system around the Sun can be neglected in a first calculation. We
can complicate things later. Take the Earth to be at the origin, and the Moon to be at a fixed
position along the x-axis.
To get to the Moon, the craft is launched from a circular orbit 50 km above the surface of the
Earth. Replace the … in the following line of code by the position from which the spacecraft is
launched. (Hint: it lies along either the x- or y-axis)
craft = sphere( pos = (...), radius = 1.0e6, color = color.red )
The craft is launched by accelerating it over a very short time to a speed of about 1.3 × 10 4 m s -1 .
5.
(a) What is the direction of this launch velocity? (Hint: it’s tangential to the initial orbit around the
Earth.)
craft.v = vector(...)
(b) Why is the launch velocity not directed straight towards the moon?
6. What is the initial momentum of the spacecraft?
craft.p = …
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III Calculations
Once launched, the craft travels under the influence of the gravitational forces of the Earth and
the Moon.
7.
(a) Write down the lines of code which calculate the gravitational forces of the Earth and the Moon on
the spacecraft.
(b) Clearly explain at least 3 differences and/or similarities between this computational model and your
mathematical model in (2c) in terms of both the code and the underlying physics.
Your lines of code should look something like this:
R = earth.pos−craft.pos
magR = mag(R)
force = −G*craft.M*earth.M*R/magR**3
Your program should solve the equations of motion of the craft in the same fashion as the “earth
and sun” program; only the craft’s motion needs to be calculated.
8.
(a) Write down the lines of code which update the position and momentum of the spacecraft.
(b) Clearly explain at least 3 differences and/or similarities between this computational model and your
mathematical model in (3) in terms of both the code and the underlying physics.
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We also need to specify a time interval between "snapshots." We’ll call this very short time
interval deltat (∆t).
To keep track of the total time elapsed, let’s also set a "stopwatch" time, t, to start from zero.
Start with a time interval of 10 s.
9. Estimate roughly how far the spacecraft travels in one time interval. Explain clearly how you arrived
at your answer.
Update the trail so that you can see the craft’s path.
You can add a rate(50) statement in the loop to limit the update rate to 50 frames per second.
You might like to put a statement scene.autoscale=0 before the start of the loop, to limit the
disturbing rescaling of the view window.
STOP ! … call a demonstrator who must check your code and sign here ___________ .
10.
(a) Sketch, with appropriate labels, what you observe in the visual output window when you run your
program.
(b) Describe your observation.
(c) Clearly explain your observation in relation to your code.
You may have to adjust the initial momentum and position of the spacecraft until its trajectory
takes it to the moon. This should require only a few adjustments.
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11. Write down the initial conditions required to get the spacecraft to the surface of the moon.
It is probably useful to check at each time step (i.e. in the loop!) whether or not your spacecraft
has reached its intended destination. The following code fragment is helpful.
Note that we use a conditional (if) statement to make the decision. We also use a print statement
to print information in the output window. Finally, we use a break statement to “break out of” the
loop and exit the program. The code assumes you have calculated the elapsed time and called it t.
rmag = mag(earth.pos−craft.pos)
# distance craft to centre of earth
if rmag

(c) Compare the visual output you observe to your prediction in (1). Clearly explain at least 3
differences and/or similarities between the visual output of your program and your prediction, in terms
of :
(i) the code, and
(ii) the underlying physics.
13. Use your program to determine the length of the voyage. Does the time calculated by your program
agree with your prediction in question 4? Clearly explain why or why not.
Part C … thinking about the energy of the system
IV More Calculations
IV.1 Work
14. Is the work done on the spacecraft during its flight positive or negative? Explain clearly.
15. Write down a mathematical expression describing the work done on the spacecraft during its flight.
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Given two vectors a and b the dot product is given by a function dot(a,b).
The force and the displacement are calculated in the loop, at each time step. The position update
statement can be broken down as follows:
deltar = (craft.p/craft.M)*dt
craft.pos = craft.pos+deltar
and deltar can be used to calculate the work done in each time interval. The total work done is
then just the sum of work done in each time interval.
This is known as numerical integration.
work = 0.0
while ...
...
w = dot(...)
work = work+w
# i.e. replace work with old value + w
Print out the value of work (together with its units!) when the voyage is over.
16. What is the calculated value of the work done? Is it positive or negative? Does this agree with your
prediction in question 14? Explain why or why not.
17. What forces do the work?
18. In your model, what is the system? (And what are the surroundings?)
IV.2 Energy
19. Write down the lines of code which will allow you to determine the change in kinetic energy of the
spacecraft during its flight.
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Make the changes to your code and run your program.
20. What is the change in kinetic energy of the spacecraft during its flight? How does it relate to the
work done on the spacecraft?
21. If you took the “whole problem” as a single system, what should be considered as part of this
system?
22. The energy of this new system should also be conserved. Is this the case here? Explain.
23. What are the three interactions contributing to the potential energy of the new system?
Part D … optional part
Now include the Sun in your program, (you have all the data you need in your original ‘Earth-Sun’
program), and let everything move. . .
24.
(a) Sketch what you observe in the visual output window when the Sun is included in the model, and
the Earth, Moon and Sun are allowed to move.
(b) Describe and explain your observation.
Get a demonstrator to check your answers and sign here ___________ before you leave.
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