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Practice questions for Physics 200, 2014 edition
Week 1, PS
Space-time diagrams soln-1.pdf
A space-time diagram is one where one axis is a direction (say, the x-direction) and one axis is time. We will put the x-direction on the horizontal axis and time on the vertical axis. The motion of an object can be represented on this diagram with a line. In this question, you will draw a space-time diagram for two children and a ball they are rolling between them. You might find it easier to do this using graph paper. Don’t forget to label everything.
(a) Jacob and Emily sit 6 meters apart. Let’s take Jacob to be at x = 0 and Emily at x=6m. At t=0, Jacob pushes a ball towards Emily with speed of 3m/s and Emily catches it once it reaches her. (i) Plot the trajectory of the ball on a space-time diagram from t=0 to t=2. (ii) As soon as she receives the ball, Emily pushes the ball back to Jacob at 2m/s, and Jacob catches it. On the same plot as before, continue plotting the trajectory of the ball. (iii) As soon as the ball left her hands, Emily turned around and crawled away from Jacob for 3 seconds at 1m/s. She then sat down again and remained sitting after that. Add a trajectory for Emily from t=0s to t=12s. (iv) After catching the ball, Jacob holds onto it for 3 seconds and then pushes it back towards Emily at 3m/s. Emily catches the ball and holds it after that. Continue plotting the trajectory of the ball until t=12s. Also, add a trajectory for Jacob, who was sitting in place the whole time.
(b) Anne cruises by on her skateboard moving with speed 1m/s in the negative x di- rection. Redraw the whole diagram from part (a) in Anne’s reference frame. Assume that Anne was at x=0 at t=0.
Drone ship explosion soln-2.pdf
When solving this question, assume that all physics is non-relativistic: everything is in- variant under the Galilean transformation and light moves with speed c with respect to an Ether. Assume the Earth and the alien planet are motionless with respect to the Ether.
A light year is the distance that light travels in one year. If you keep all your distances
in light years, it will be much easier to deal with the numbers. For example, a ship travelling at 0.8c needs 12.5 years to travel 10 light years.
An alien civilization from a solar system 210 light years away from ours decides to get in touch with us. At the same time, they send a message by laser, a drone spaceship trav- elling with speed 0.8c and a larger, fully crewed spaceship travelling with speed 0.2c. A political faction opposed to the mission puts a bomb on the drone ship and sets its timer to 15 years.
a) What gets to Earth faster: the message sent by laser, or the light from the explo- sion of the drone ship? Justify your answer carefully.
b) When the light from the explosion reaches the second ship, the crew decides to send another drone to replace the first. They would like it to reach the Earth at the same time as the first ship would have if it were not destroyed. How fast, relative to their own ship, must the crew launch the new drone ship?
Week 2, PS
Trains passing soln-3.pdf
Two trains, A and B, are moving towards each other with relative speed 0.8c. The passing of the two trains (from when their fronts align to when their backs align) takes 20% longer as observed from train A as is does as observed from train B. What is the ratio of the length of A to the length of B?
Time dilation soln-4.pdf
Maggie, eleven years old, is very jealous of her younger brother Jacob. Jacob, at ten years younger, is just a baby, and Maggie feels that he gets all the attention at home. She sneaks onto a spaceship leaving for Barnard’s Star, a very low-mass red dwarf star 5.96 light years away from the solar system. She is hoping that by the time the ship gets back, she will be the baby of the family. Her plan works and when she returns, she is (biologically) a year younger than Jacob. How old is Jacob when Maggie gets back from her trip? Assume that
swers in m/c (meters divided by the speed of light c).
(b) Analize this question in the frame of reference of the moving train.
(c) In the frame of reference of the ground, how far apart do the two firings occur?
(d) Write the Lorentz tranformation which ties together your answers from parts (a) and (b). Confirm that it is satisfied.
Missing Jar-Jar soln-8.pdf
A Rebel Alliance starship is cruising at v = (3/5)c at an altitude of 10km above flat ground. The ship’s laser gun is jammed so that it points exactly downwards, perpendicular (in the ship’s frame) to the direction of motion of the ship. Jar Jar Binks is standing on the ground. When a ship is directly over Jar Jar, the pilot fires the gun. The laser hits a nearby tree.
(a) How far away is was the tree from Jar Jar?
(b) In the ground frame of reference, what are the horizontal and vertical components of the velocity of the laser pulse?
Magical rulers soln-9.pdf
We will re-examine question 3 from Tutorial 2. Two parallel rulers, one (ruler A) stationary and one (ruler B) moving with velocity v =
p 3 c/2 in the positive x direction, appear to be the same length, L. (See picture in the Tutorial sheet.) Define two events:
I - the left ends of the rulers line up II - the right ends of the rulers line up
(a) In the frame in which ruler A is stationary, let event I be at (t, x) = (0, 0). What are the space-time coordinates of event II in this frame? (b) Use Lorentz transformations to figure out the space-time coordinates of the two events I and II in the frame in which ruler B is stationary. (c) How long does it take, in the frame in which ruler B is stationary, between event I and event II?
Now that you have solved the problem, you might want to compare with solutions to the Tutorial.
Supernova half-way soln-10.pdf
A spaceship traveling with speed v = c/2 leaves the Earth for a star ten light years away. You can assume that the star is stationary w.r.t. the Earth. When the ship is half way to its destination, in the Earth’s frame of reference, the star goes supernova (blows up).
(a) In the ship’s frame of reference: how far from the Earth is the ship when the su- pernova occurs? (Events simultaneous in the ship’s frame of reference.)
(b) How much time passes in the spaceship’s frame of reference between the ship leaving the Earth and the light from the supernova explosion seen on the ship?
Challenge Re-do the question with v = (
p 3 /2)c.
Week 4, PS
Joker and Batman soln-11.pdf
Joker and Batman are driving along a long straight road. While passing a tree, Joker sets it on fire (because the visual e↵ects would be cool). Batman is approaching the tree with speed 0.6c; in Batman’s frame of reference, Joker is exactly halfway between him and the tree when the first light from the fire reaches Batman’s car.
(a) How fast is Joker driving (with respect to the ground)?
(b) 1 μs passes in the Joker’s frame of reference between the moment Joker sets the tree on fire and the moment he and Batman crash into each other. In the ground’s frame of reference, how far away is Batman from the tree when Joker sets it on fire?
Harry Potter soln-12.pdf
in touch with us. At the same time, they send a message by laser, a drone spaceship trav- elling with speed 0.8c and a larger, fully crewed spaceship travelling with speed 0.2c. A political faction opposed to the mission puts a bomb on the drone ship and sets its timer to 15 years. When the light from the explosion reaches the second ship, the crew decides to send another drone to replace the first. They would like it to reach the Earth at the same time as the first ship would have if it were not destroyed. How fast, relative to their own ship, must the crew launch the new drone ship?
To get the most practice from this question, you can consider doing it in each of the three frames of reference!
Week 6, PS
Proton-proton collision soln-15.pdf
Two protons collide at the Large Hadron Collider (LHC), each traveling at a velocity u such that (u) = 7 · 10 3. A proton weighs about 1GeV/c 2.
(a) How fast are the protons moving?
(b) The two protons form an unstable particle. What is this particle’s mass?
(c) The unstable particle decays to form two hypothetical proton-like particles each with rest mass 1000GeV/c^2. How fast are these hypothetical particles going?
(d) Is it possible to instead produce a di↵erent hypothetical particle with rest mass M = 16 TeV/c 2 in this collision?
Unstable particle soln-16.pdf
An accelerator experiment produces a new unstable particle X at rest. X quickly decays to a tau (mass 1777M eV /c 2 ) and a tau anti-neutrino (which we will take to be massless). Neu- trinos are very di cult to observe, so its energy cannot be measured. The tau is measured to have energy of 4000M eV. What was the mass of the mystery X particle?
Energy production in the Sun soln-16A.pdf
The Sun’s total luminosity is 3. 8 ⇥ 10 26 W.
(a) Assuming the luminosity is a constant, by how much will the Sun’s mass decrease in the next 2 billion years?
(b) Assuming that the dominating source of the Sun’s energy is conversion of four protons and two electrons into one helium-4 nucleus and two neutrinos, and that all of this energy is lost via radiation (and none is lost to neutrinos), how many moles of hy- drogen does the Sun convert to helium in one second? Masses: m(p +^ ) = 938. 272 M eV /c 2 , m( 4 He +2^ ) = 3727. 379 M eV /c 2 , m(e ^ ) = 0. 510 M eV /c 2 , m(⌫) ⇡ 0 M eV /c 2.
E↵ective mass soln-17.pdf Consider Newton’s second law in relativistic setting:
F^ ~ = d~p dt
d dt
m~u q 1 |~u| 2 /c 2
Consider a particle moving in the x-direction on which a force acts in the y-direction. This causes the y-component of ~u to change, while the x-component stays constant, u (^) x = U. At t=0, u (^) y = 0.
(a) Compute the y-component of the acceleration at the instant t = 0 in terms of F , m and (U ).
(b) If you wanted to write a formula of the form F = M a based on your answer to part (a), what would M have to be?
Relativity and EM soln-18.pdf
- This question is a bit harder than the typical homework problem in this course. ** In this question you will verify that length contraction together with time dilation resolves the discrepancy you saw in Tutorial 1. You may use the following facts about electromagnetic forces:
- An infinitely long line of charges with charge density ⇢ exerts a force ⇢dQ on a charge Q a distance d away.
Before After
0.6c 0.8c
(a) What are the energies of the two photons A and B? Give answers in M eV /c 2 and make it clear which photon has which energy.
(b) Why is it not possible for the photons to be produced moving along the y-axis, as shown below?
Before After
0.6c 0.8c
(c) If just one of the two photons is produced moving along the y-axis, in what direc- tion must the other photon be moving (compute the angle)?
Ship propelled by a laser soln-20.pdf
In this question, we will consider a Sci-Fi concept, a ship driven by an Earth-based laser. This works as follows: a really powerful and well collimated laser beam is sent from Earth in the direction the ship is supposed to go in. The spaceship is equipped with a large mirror and the laser beam is reflected back to the Earth. Since light carries momentum, this propels the ship forward. Let’s consider a time after the initial boost-o↵, when the laser is o↵ and ship is moving with a large velocity v away from the Earth. To give the ship’s velocity a final adjustment, a single short laser burst is emitted towards the ship, with total energy E in the Earth’s frame of reference. How much momentum, in the Earth’s frame of reference, does this burst give to the ship once it’s been reflected o↵ its mirror?
Hint I: You may assume that the spaceship is very massive and therefore, in the ship’s frame of reference the momentum of the reflected pulse has the same magnitude as the momentum of the incident pulse. Equivalently, in the ship’s frame of reference, the reflected light has the same wavelength as the incident light. Hint II: There are two ways to do this problem. You can either use Lorentz transforma- tions for p and E or the Doppler shift formula you derived in Tutorial 6.
A challenging extension (not for credit) A more realistic version of this problem would have a continuous laser beam delivering the momentum to the ship over a long period of time. To figure out the motion of the ship if the laser beam is continuously turned on, prove that
dv dt = 2(P/M c)(1 + v/c) 1 /^2 (1 v/c) 5 /^2
where P is the total power of the laser beam, M is the mass of the ship and t is time in the Earth’s frame of reference. Then try to integrate to get the velocity as a function of time.
Bullet striking a toy car soln-21.pdf
A bullet, moving with speed 0.8c and with mass 5g strikes a motionless toy car weight- ing 50g. What is the speed of the toy car after the bullet wedges in it and how much mass the toy car with the bullet wedged in it have?
Two bullets colliding soln-22.pdf
Two bullets collide head on. The first bullet is going with speed 0.1c and weights 5g. The second bullet is going with speed 0.2c and weights 3g. Assume that after the collision, the two bullets form one stuck-together mass.
(a) What is the mass and the velocity of the resulting object?
(b) Assume that before they collided, the bullets were at a temperature of 200C. The heat capacity of lead is 130 J/(g C). If the increase in rest mass energy is due to heat, how hot is the resulting object? (assume that heat capacity per atom is constant at all temperatures, and make other simplifying assumptions: ignore that lead will melt, ignore that atoms will form plasma, etc...)
2(a) A photon with wavelength 500nm scatters o↵ a stationary electron. If its new direction of motion is perpendicular to the original direction, by how much does the wave- length of the photon change? Based on this number, can you explain why the Compton e↵ect is not observed with visible light?
2(b) A photon with wavelength scatters o↵ a free electron.
(i) At what angle of scattering is the wavelength of the photon changed the most?
(ii) In a Compton scattering experiment, some photons were seen to have their wave- length changed by as much as 1%. What is the maximum wavelength of the incoming photons?
Photoelectric e↵ect symulation soln-26.pdf
In this problem, you will use the photoelectric e↵ect simulation again. You can find the simulation at http://phet.colorado.edu/en/simulation/photoelectric Select the Mystery Metal (?????) as your Target. Measure and plot the stopping potential at di↵erent light frequencies. From your plot, determine:
(a) The work function of the Mystery Metal.
(b) An approximate value for Planck’s constant, given that the electron change is e =
- 6 ⇥ 10 ^19 C.
Make sure to label your axes, as well as explain what you did and why!
Black body radiation soln-27.pdf
The qualitative features of black body radiation are described by two laws:
- Stefan’s law says that the power radiated per unit area of the body is given by I = T 4 where = 5. 67 ⇥ 10 ^8 W/m 2 K 2 is the Stefan-Boltzmann constant
- Wien’s law gives the wavelength at the peak to be
(^) peak =
- 90 ⇥ 10 6 nm K T
Make the approximation that most of the radiation has wavelength close to (^) peak. If you double the temperature, how does the number of photons radiated per second change?
Week 9, PS
Trig identities made simple soln-28.pdf
Complex numbers are useful for many things. In this question, you will derive some trigono- metric identities using the power of complex numbers.
(a) Consider the following fact:
e i(x+y)^ = (e ix^ )(e iy^ )
with x and y real. Write e i(x+y)^ , e ix^ and e ix^ in the form a + ib, then multiply out the RHS. By considering the real and imaginary parts of the resulting equation, derive formulas for cos(x + y) and sin(x + y).
(b) Now, consider this equation:
e 4 ix^ =
⇣ e ix^
⌘ (^4)
Use it to prove that sin(4x) = 4 cos 3 (x) sin(x) 4 cos(x) sin 3 (x).
Di↵raction pattern soln-29.pdf
The attached figure (on the next page) shows a di↵raction pattern (intensity in some ar- bitrary units as a function of distance on the screen) for a two-slit experiment. Using the figure, estimate as accurately as you can the probability that the first two photons both hit the screen within the central interference fringe.
Polarizer at 60-degrees soln-30.pdf
Consider a polarizer with its axis pointing at an angle of 60 ^ to the y direction (vertical direc-
tion), as shown.
(a) What polarization must incoming light have to be completely transmitted through the polarizer? (Write your answer in the form E (^) x ˆx + E (^) y yˆ. You should be able to determine E (^) x and E (^) y up to an overall normalization). What polarization must incoming light have to be completely absorbed by the polarizer?
(b) Let ˆe (^) t be a unit vector pointing in the same direction as the transmitted light’s polarization, and let ˆe (^) a be a unit vector pointing in the same direction as the absorbed light’s polarization. What are ˆe (^) t and ˆe (^) a?
(c) Solve for ˆx and ˆy in terms of ˆe (^) t and ˆe (^) a.
(d) Now consider elliptically polarized light with polarization E(1 i)ˆx + Eiyˆ. Write its polarization in terms of ˆe (^) t and ˆe (^) a (using your answer form part (c)). What fraction of this light goes through the polarizer?
Same question, new notation soln-31.pdf
In this question, we will redo the previous problem, using the photon picture. If you did the previous question, this one is really just an exercise in notation.
(a) With the same polarizer set-up as in Question 2, write down a quantum state for a photon that will be transmitted by the polarizer with 100% certainity, and the state for a photon that will be absorbed with 100% certainity. Give your answers as quantum superpo- sitions of | 0 i and | 90 i and don’t forget to normalize properly.
(b) Let’s denote the transmitted photon and absorbed photons from part (a) by |ti and |ai, respectively. Solve for | 0 i and | 90 i in terms of |ti and |ai.
(c) Now consider a photon from a light beam with polarization E(1 i)ˆx+Eiyˆ, same as in Question 1(d). What is the quantum state of this photon, and what is the probability of this photon going through the polarizer? (Don’t use your answer from Question 2(d). Instead, redo your computation from that question using the photon quantum state notation.)
Circular polarization soln-32.pdf
An elliptically polarized photon might have a quantum superposition with complex coef- ficients.
(a) Consider photons described by (| 0 i + i| 90 i)/
p 2 and (| 0 i i| 90 i)/
p
- Explain why these are called right and left circularly polarized photons [Hint: review the tutorial.]
(b) What is the probability that a photon in a quantum state (| 0 i + i| 90 i)/
p 2 will go through a vertically oriented polarizer?
(c) What is the probability that this photon will go through a polarizer at an arbi- trary angle ✓?
Challenge question (not for credit) In the case of the circularly polarized photons from the practice problem above, describe how you would measure the polarization of a beam of such photons if you had a polarizer and a quarter-wave plate.
Week 11, PS
A constant wavefunction soln-33.pdf
Consider a particle with the following wavefunction:
(x) =
( A for |x| < d 0 for |x| > d (a) Find a value for A which makes the wavefunction normalized.
Question 1(c), sketch and describe what will be visible on the detector screen if many such electrons were to pass through the system and hit the screen. (c) Describe how you could determine the half-width of the slit d from the experimental data consisting of the positions of the fringes, the velocity of the electrons v, the mass of the electron m (^) e and the distance D. (d) If the slit is too narrow, its width cannot be determined. Let’s take the total size of the projector screen to be D. What is the smallest half-width d of the slit that can be measured using this method? (e) Compare your answer to part (d) with the de Broglie wavelength of the incident electrons.
Intensity profile and probabilities soln-35.pdf
Light passes through one slot on a way to a screen. The intensity profile on the screen is measured to be I (^0) cosh(x)
where I 0 is the intensity at the center of the pattern.
(a) What is the probability that a given photon will hit the screen between x = 0 and x = 1?
(b) What is the probability that the first four photons will hit the screen at x > 0?
Remark 1: The hyperbolic cosine, cosh(x), is defined as (e x^ + e x^ )/2. Remark 2: The above function does not integrate to 1. Remark 3: If you are having trouble with the integration, the following substitution might help: e x^ = tan(y).
Two slit interference and resolution limit soln-36.pdf
You might be familiar with the fact that the resolution of a microscope is ultimately limited by the wavelength of the light used. In this question, we will examine this. We know that when light passes through small features (such as narrow slots), instead of getting the image of the features on the screen, we get interference fringes. Let’s examine how close together two really narrow slits can be before we can no longer resolve them (see that there are two slits and not just one).
(a) Explain, using the Figure as your guide, why the double slit interference pattern gets wider when we make the slits closer together.
(b) Sketch what the pattern looks like on the screen if the first dark fringe is at ✓ = ⇡/2. Can you infer the existence of two slits (as opposed to one slit twice as wide) from this pattern?
(c) For light with wavelength , what is the distance D between the slits when ✓ = ⇡/2?
