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Physics 118 – Exam 3 Apr 03 – 04, 2020

Honor pledge: All work presented here is my own (sign) ____________________________ • This exam consists of 6 questions worth 100 points. It you print the exam, make sure you

have printed all the pages. If you work on your own paper, make sure to include exactly 1 page per question plus one page for the cover sheet with your honor code signature.

• Write all work within the page borders to prevent losing work when the exams are scanned!

• The equation sheets are the last 2 pages of the exam. Please do not scan them when you return the exam.

• You have 90 minutes to complete the exam; you will be allowed an additional 30 minutes for downloading, printing, scanning, and uploading the exam, however, we expect that you only spend 90 minutes (self-timed) working on the actual exam. Working beyond this time limit would constitute an honor code violation.

• This is a closed book exam. Only writing utensils and a calculator are allowed on your desk.

• The calculator must abide by the calculator policy (no graphing or solving calculators). The calculator must not be contained in a cell phone or other internet capable device.

• Please write neatly and legibly. If you need more space, use a separate page and include it at the end of your scanned exam – make sure to indicate a separate page is being used on the main part of the problem.

• Unless stated otherwise, you may assume that the speed of light is 3.0 × 108 m/s.

• Numerical answers should be rounded to an appropriate number of significant digits.

• To receive full credit, you must show all work, unless stated otherwise. Provide enough detail in your answers so that the grader can follow your reasoning. Clearly identify your final answer for each problem.

• Exams can be stressful – please remember this is only a portion of your final course grade. Take a moment to remind yourself of something you enjoy, relax, and do your best!

Please scan and upload this cover page as part of your Gradescope submission. Total number of pages is 8 – cover page plus 6 questions over 7 pages

Name (Print):

PID: Studio:

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Q1. (20 pts). A scientific spacecraft is traveling at a constant speed of 0.95c away from the Earth. The spacecraft eventually passes the asteroid belt at a distance 630 million km from the Earth (as measured in the reference frame of the Earth). Assume that the asteroid belt is at rest with respect to the Earth. The spacecraft emits a communication laser pulse toward the Earth with duration of 2.0 x 10-6 s (as measured by the spacecraft). (a) What is the duration of the pulse that observers on the Earth receive, in seconds?

(b) How much time does it take, in the reference frame of the Earth, for the laser light to travel

from the spacecraft to the Earth? Please provide your answer in minutes.

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Q2. (20 pts). Consider a rod at rest in reference frame S that has an angle θ from the x-axis as shown in the figure below. An observer in an inertial reference frame S’ moving at a positive x velocity looks at the rod and measures its angle θ’ from their x’ axis.

a) Is θ’ greater than, less than, or equal to θ? Explain your reasoning.

b) Now consider a new reference frame S’’ moving with a negative x velocity relative to S. An observer in that frame measures the rod angle with respect to the x’’ axis as θ’’. How does θ’’ compare to θ’, the angle measured by S’ when moving with positive x velocity? Explain your reasoning.

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Q3. (20 pts). The spacetime diagram to the right, in frame S, depicts four events labeled A, B, C, and D. The grid spacing in each direction is equal to 1 light year (1 ly). Answer the questions related to this diagram in the space below. a) Is it possible for event D to cause event A? Why or

why not? b) Is it possible for events B and C to be simultaneous in a different reference frame? Why or

why not? c) What is the space-time interval s2 between events A and C? Now, consider a reference frame S’ in which events A and C are simultaneous. d) What is the space-time interval s2 between A and C in S’? e) In what direction is the velocity of frame S’ with respect to S? Justify your answer.

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Q4. (15 pts). Consider two events, A and B. In an inertial reference frame S, event A occurs at a time t after event B, and event A occurs at position x = 0 and event B at x = L. From another reference frame S’ it was observed that events A and B occur simultaneously. Given this information, what is the relative velocity of S’ to S? Express your answer in terms of the speed of light c, and the parameters given in the problem. You will not be graded on a spacetime diagram, but we strongly suggest that you draw one to help you solve the problem.

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Q5. (10 pts). Admiral Picard is on a space station, and he sees Captain Riker fly by in a spaceship moving with speed V. Picard flashes a red light, waits one second as measured by his watch, and then flashes a blue light. Next to each of the following statements, indicate, or write in, whether the statement is true or false. No explanation is needed. a) Picard measures the light from the flashes to be moving at the same speed that Riker measures. TRUE / FALSE b) The two flashes occur at the same location in Riker’s reference frame. TRUE / FALSE c) It is possible that the blue flash occurs before the red flash in Riker’s reference frame. TRUE / FALSE d) In Riker’s reference frame, the speed of the space station is V. TRUE / FALSE e) In Riker’s reference frame, the time between the flashes is longer than one second. TRUE / FALSE

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Q6. (15 pts). An exploration ship travels from our sun (Sol) toward Alpha Centauri at a constant velocity. In the reference frame of the ship, when the ship is halfway between the stars, it emits a radio signal back toward Sol that indicates the ship is having an unknown problem. Upon receiving the signal, the people of the Sol system immediately launch a second ship at the same velocity toward Alpha Centauri. The second ship arrives at Alpha Centauri and locates the first ship, empty of its passengers, with a single, mysterious message “CROATOAN” displayed on their communication screens. Draw a space-time diagram for this scenario in a reference frame moving at the same velocity as the ships when they are in transit between the star systems. Set the origin of your diagram to coincide with the departure of the first ship from Sol, and label your axes. Your diagram will be qualitative, however, it must be readable when scanned. On your diagram, clearly indicate each of the following using the given letters:

A. The first ship’s world line, assuming it continues on a straight path toward Alpha Centauri at constant velocity

B. The worldline of Sol C. The worldline of Alpha Centauri D. The world line of the radio signal E. The second ship’s world line F. The event where the first ship departs from Sol G. The event where the radio signal is emitted from the first ship H. The event where the radio signal is detected at Sol & the second ship departs I. The event where the second ship arrives at Alpha Centauri

Make the essential part of your diagram as large as possible on the page. Use the letters of the items listed above to indicate them on your diagram (for example, you should use “A” to indicate the world line of the first ship). You might wish to make a test drawing on scratch paper first, to plan out how to make it fit comfortably on one page.

Make your drawing on the page immediately following this page. Do not draw anything here.

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Drawing for Question 6:

Physics 118 Exam Equation Sheet

Factor

Symbol

prefix

Conversion Factors, Formulas, and Constants 1012

T

tera-

1 lb. = 4.45 N

109

G

giga-

1 in. = 2.54 cm

1 m = 3.28 ft.

1 ft. = 0.305 m 106

M

mega-

1 mile = 5280 ft.

REarth = 6371 km

103

k

kilo-

c = 3×108 m/s

MEarth = 5.97×1024 kg 10–2

c

centi-

gEarth = 9.81 m/s2 = 32 ft/s2 10–3

m

milli-

G = 6.674×10–11 m3/kg/s2

10–6

µ

micro-

C = 2πr REarth-Moon = 384,400 km 10–9

n

nano-

Acircle = πr2

10–12

p

pico-

Asphere = 4πr2 Vcylinder = πr2h 10–15

f

femto-

Vsphere = (4/3)πr3 ρ = M/V

s (meters) = rθ

vt = rω ω = 2πf = 2π/T ω (radians/sec) =

α (radians/sec2) =

K = ½Iω2 Rolling: vCoM = rω

Δr = rf −

ri!r(t)− !r0 =

!v0t +12!at2

!v = d!rdt

- D= =

- D

!!

! !final initial

averagefinal initial

r rvt tr

t!v(t)− !v0 =

!at

!a = d!vdt

!aaverage =!vfinal −

!vinitialt final − tinitial

= Δ!v

Δtvi2(t)− vi02 =2ai(ri(t)− ri0)i = x , y ,z

R = 2v02 cosθ sinθ

g= v0

2 sin2θg

H = v02 sin2θ2g

T = 2v0 sinθg

θ(t)−θ0 =ω0t +12αt

2

dθdt

ac =v2tr= rω 2 ω −ω0 =αt

dωdt

at = rα ω 2 −ω02 =2α(θ −θ0)

!p =m!v !J =

!Fnet ,avgΔt = Δ

!p!F =!Fnet = m∑ !a = d

!pdt

!F12 = −

!F21 fs ≤ µsFnormal fk = µk Fnormal

!Fspring = −k(Δ

!s ) ΔK =Work =!F id!r =∫ cosθF(θ ,r)dr =∫ Fxdx +Fydy +Fzdz( )∫

ΔK +ΔU +ΔEthermal =Workexternal K = 12mv

2 = 12p2

mD = D 2

1 ( )2springU k s ΔUgravity =mgΔh

!F = −G mM

r2r̂ U = −G

mMr T

2 = 4π2

GM⎛⎝⎜

⎞⎠⎟r3

Physics 118 Exam Equation Sheet

1 light-year = c(1 yr) = 9.461×1015 m

p = γmv E = γmc2 K = (γ–1)mc2

E2 = p2c2 + m2c4

SHO:

Physical Pendulum:

Parallel Axis Theorem: Id = ICoM + md2

y(x, t) = Acos(kx ± ωt)

Classical Doppler Effect:

Speed of sound: 331 m/s @ STP

343 m/s @ 20 °C

Relativistic Doppler Effect:

′x = γ (x −βct)= γ (x − vt) c ′t = γ (ct −βx)= γ (ct − v

cx) s

2 = (cΔt)2 −(Δx)2 = (cΔ ′t )2 −(Δ ′x )2

x = γ ( ′x +βc ′t )= γ ( ′x + v ′t ) ct = γ (c ′t +β ′x )= γ (c ′t + v

c′x )

β = v

c γ = 1

1−β 2β 2 =1− 1

γ 2 1+α( )n≈1+nα forα