SAMPLE EXAM FAMILY ARCHIVE---RELATIVITY,  WAVES: TRAVELING AND STANDING WAVES; INTERFERENCE

RELATIVITY: PHYSICS 5  SP '12 REFERENCE PROBLEMS: QUIZZES 1,2, CH. 37; hints at www.nvaphysics.com.

4C FINAL FALL’99: LENGTH CONTRACTION

1. (20  points)

An observer, moving at a speed of 0.900c relative to a rod,  measures its length to be
1.5 m and sees its length to be oriented at 25-degrees with respect to the direction of motion.

(a) (10 points) What is the proper length of the rod as measured in the reference
                        frame moving with the rod?

(b) (10 points) What is the orientation angle in the reference frame moving with the

                        rod?

 

 

HINT: ONLY THE DIMENSIONS PARALLEL TO MOTION ARE CHANGED; THUS THE HORIZONTAL LEG OF THE ANGLE WILL ONLY BE CHANGED,  NOT THE “OPPOSITE” SIDE PERPENDICULAR TO MOTION.

4C FINAL FALL ‘98:  RELATIVE VELOCITIES: QUIZ 2, #22, 23, 59

2.( 20  points)

 

Galaxy  A  is reported to be receding from Our Galaxy with a speed  of 0.25c. Galaxy B is located precisely in the opposite direction. Galaxy B if found to be receding from Our Galaxy at a speed of 0.45c.

 

(a) (6 points) What recessional speed would an observer on Galaxy A find for
                      Our Galaxy?

 

(b) (14 points) What recessional speed would an observer on Galaxy A find for
                         Galaxy B?

 

 

4C FINAL SP ’00 (Y2K SPECIAL!): SEE QUIZ 2, #58 DISCUSSION COMMENTS AT www.nvaphysics.com.

3. (20 points)  A pion at rest ()decays into a muon ()and a photon (m = 0). See the diagram of the reaction below. Find the speed v of the muon after the reaction. You may use the conversion factor

 


Hint: You have two equations and two unknowns, the muon’s speed v and the photon’s energy = P*c, where P is the photon momentum; so really the photon’s momentum P is the second unknown. But we can find both  quantities  using two equations~ conservation of energy and conservation of momentum:

1. Mpion*c2 = P*c + Mmuon*c2/(1 – v2/c2)1/2  .
2. P = Mmuon*v/(1 – v2/c2)1/2  .
Solve these two equations for v. It is best to substitute equation 2. into 1. and  get v by clearing the fractions and raising both sides of the equation to the appropriate  power.

 

FINAL 4C SP ‘01  RELATIVE VELOCITIES: QUIZ 2, #22, 23, 59; ALSO TIME DILATION , LENGTH CONTRACTION; RELATIVISTIC MOMENTUM

4. (20 POINTS) A space ship whose rest length is 200 m is approaching the earth at a speed of 0.30 c with respect to the earth.    Note: 200 m is the length of the ship as measured by a passenger on the ship.   A meteorite, with speed 0.90c with respect to the earth, is in front of the ship moving in the opposite direction. It passes the ship on the parallel track.

(a) (5 points) What is the length (in meters) of the spaceship as measured by  a person on the Earth ?

(b) (6 points) What is the length (in meters) of the spaceship as measured by an alien being on the meteorite.
(c)( 3 points) What is the magnitude of the momentum of the meteorite measured by a passenger on the spaceship if the rest mass of the meteorite is m = 1.0 kg ?

(d) (3 points) How long (in seconds) does it take the meteorite to pass the ship as measured by the alien on the meteorite ?
(e) (3 points) How long (in seconds) does it take the meteorite to pass the ship as measured by a passenger on the ship ?

 

FINAL 4C SP ‘ 02 RELATIVE VELOCITIES: QUIZ 2, #22, 23, 59; RELATIVISTIC MOMENTUM; LENGTH CONTRACTION.

5. (30) Two small rockets A and B fly away from each other each other at speed 0.75 c relative to the earth. The rockets each have mass m = 2000.0 kg

(a) (10) What is the magnitude of the momentum of  A relative to B?
(b) (10) What is the magnitude of the momentum of  B relative to A?

(c) (8) Suppose the length of the rocket B relative to the B reference frame is
200 m. (This is called the proper length.) What is the length of rocket B relative to the A-reference frame? Relative to the earth ?

(d) (2) circle the correct answer: Who invented the special theory of relativity?

   (1) Lorentz  (2) Einstein   (3) Hertz  (4) Maxwell

FINAL 4C SP ‘04 RELATIVE VELOCITIES: QUIZ 2, #22, 23, 59; RELATIVISTIC MOMENTUM; LENGTH CONTRACTION.

6. (22   points)   A reconnaissance spaceship (A) is moving toward your spaceship (B).  The reconnaissance ship (A) fires a friendly missile along a path parallel to your line of motion with speed 0.700c relative to the frame of the spaceship A. You, an observer on spaceship B, measures that the missile is approaching you with speed 0.960c. The spaceship A  has length L0 = 30.00 m as measured in the A’s  frame of reference.  (This is called the proper length.)

 

 

(a)    (9)What is the speed of spaceship A relative to you (B) ? 

(b)    (9)What is the length of spaceship A as measured by you (B)?

(c)    (4)Assume the missile has mass 3.00 kg. What is the magnitude of the momentum of the missile as measured by you (B)?

 

 

 

FINAL 2B SP ‘11 RELATIVE VELOCITIES: QUIZ 2, #22, 23, 59

7. ( 20 points) A United Nations observer on Planet  Earth (1) remotely views  two vehicles in space.  A small rocket (3) has been  fired from a large rocket (2).  Rocket 3’s  velocity  relative to rocket 2  is V32 = 0.600c. Rocket 2’s velocity relative to the Earth is V21 = 0.600c. See Figure 1.

(a) (13 points) Find the  velocity V31  of  rocket 3 relative to the Earth.

(b) (2 points)   Explain why the answer to part (a) should  be less than the speed of light c.

(c) (5 points)  See Figure 2 on the next page. Suppose  rocket 2 sends out a beam of light (3) . In this case  V32  =c.  Assume Rocket 2’s velocity relative to the Earth is V21 = 0.600c . Using the method and formulas you used in part (a) , compute the  velocity V31  of  rocket 3 relative to the Earth. Does your answer make sense? Explain.

 

FINAL 2B SP’ 08    RELATIVE VELOCITIES: QUIZ 2, #22, 23, 59; ALSO LENGTH CONTRACTION, TIME DILATION


8.  (40 points) A spaceship (S) moving away from the Earth (E) with  velocity  VSE = 0.95c  launches a probe (P) in the forward direction with velocity VPS = 0.20c relative to the ship. In other words,
VSE = velocity of ship relative to Earth
VPS  = velocity of probe relative to ship
(a) (15 points) Show that the velocity VPE of the probe relative to the Earth has a magnitude less than  c. 

(b) (9 points) A signal  on the probe flashes on and off once every 4.0 seconds. How much time elapses between flashes of the signal as measured by a scientist on Earth? 

(c) (8 points)  Suppose the length of the probe is 50.0 m as measured in the probe’s reference frame.  What is the length of the probe as measured by a scientist on Earth?

(d) (8 points)  Suppose the rest mass mo of the probe is 1000.0 kg. What is the magnitude of the probe’s momentum as measured by a scientist on Earth?

FINAL 2B SP’09 TIME DILATION, LENGTH CONTRACTION

9.  (12  points) Special Relativity

(a) ( 6 points)  A certain type of experimental elementary particle travels at a speed v =  2.65x10 8 m/s relative to the laboratory. At this  speed , the average lifetime of the particle is measured by a person in the laboratory to be Δt = 4.75 x10 -6 s.   What is the particle’s lifetime Δto in the particle’s reference frame? In other words, what is  particle’s lifetime Δto  in a reference frame in which the particle is at rest?

(b) ( 6 points) A spaceship passes  you at a speed of v = 0.750c. In the spaceship’s reference frame, the length of the spaceship is Lo = 43.0 m. In other words,  the spaceship has length Lo = 43.0 meters in a reference frame in which the spaceship is at rest. What length L  (in meters) do you measure the  moving spaceship’s length to be?

 

 
TRAVELING WAVES: PHYSICS 5  SP '12 REFERENCE PROBLEMS: QUIZ 3, CH. 15, #4, 6, 12, 16, 20, 24  at www..masteringphysics.com; hints at www.nvaphysics.com.

4C TEST 1 AU ‘98

10. (20 points)  A transverse wave on a string is described by:

                                       ,

where x is measured in meters.

(a) (5  points) What distance does a wave crest move in 10 seconds?
(b) (5  points) Does the crest move in the positive or negative x-direction ?
(c) (5  points) Determine the transverse velocity of  the string  at t = 0.20 seconds for the point on the string located at 1.6 m.
(d) (5  points) Determine the transverse acceleration of  the string  at t = 0.20 seconds for the point on the string located at 1.6 m. 

 

Hint: Find the transverse velocity by taking the partial derivative with respect to time.

4C TEST1 SP ‘01

11. (16 points)

 

A transverse wave on a string is described by :

(a) (4 points) Find the transverse velocity of a particle on the string at the time t = 0.20 sec and x =1.6 m.
(b) (2 points) Assume that up is in the positive y-direction and down is in the negative y-direction.  Is the particle moving up or down at the value of  t and x used to find the velocity in part (a)?

(c ) (4 points) Find the transverse acceleration of a particle on the string at the time t = 0.20 sec and x =1.6 m.
(d) (1 point) What is the wavelength ?

(e) (1 point)  What is the period ?

(f) (1 point)  What is the speed of the wave in the x-direction ?

(g) (3 points) Is the wave moving in the positive or negative x-direction?

4C TEST 1 SP ‘02

12. (30 points) Ocean waves  with a crest-to-crest distance of 20.0 m can be described by the equation:

y(x,t) = (0.800 m) sin ([0.628(x + vt)] where v = 1.30 m/s

 

(a)   (12) Sketch y(x,t)  at t =0.  Be careful. I’ve got to see the wavelength shown. Draw at least 2  wavelengths of the wave and show the amplitude, just like you did carefully on the homework. Use the axes (a) shown below.

(b)   (18) Sketch the wave at t = 2.0 seconds. Be careful. Show the wave directly underneath the wave drawn in part (a). Indicate how the wave at t= 2.0 seconds is shifted relative to the part (a) wave.  Use the axes (b) shown below.

4C TEST 2 SP ‘03

13.  (23 points)

 

A transverse wave on a string is described by :

(a) (4 points) Find the transverse velocity of a particle on the string at the time t = 0.20 sec and x =1.6 m.
(b) (5 points) Assume that up is in the positive y-direction and down is in the negative y-direction.  Is the particle moving up or down at the value of  t and x used to find the velocity in part (a)?

(c ) (4 points) Find the transverse acceleration of a particle on the string at the time t = 0.20 sec and x =1.6 m.
(d) (2 point) What is the wavelength ?

(e) (2 point)  What is the period ?

(f) (2 point)  What is the speed of the wave in the x-direction ?

(g) (4 points) Is the wave moving in the positive or negative x-direction?

 

STANDING WAVES: PHYSICS 5  SP '12 REFERENCE PROBLEMS: QUIZ 3, CH. 15, #38; SEE ALSO #12 FOR A  TRANSVERSE VELOCITY REVIEW  at www..masteringphysics.com; hints at www.nvaphysics.com.

4C TEST 1 SP ‘ 07

14. (Extra Credit) (23) A thin taut string tied at both ends and oscillating in its fourth harmonic has a shape given by

y(x,t) = (8.40 cm) sin[(0.440π rad/cm)x]sin[(50.0π rad/s)t].

The origin is at the left end of the string and the x- axis is along the string. The y-axis is perpendicular to the string. Assume right is the positive x-direction. Assume up is the positive y-direction.

 

(a) (2)What is the length L of the string?
(b) (2) Sketch the standing wave pattern between x = 0 and x = L, inclusive.
(c) (2)Find the amplitude A of each of the two traveling waves making up the wave.
(d) (4)Find the maximum transverse speed |vy| of a point on the string?
(e) (4) At what values of x does the maximum transverse speed occur?
(f)  (4)At the position x = 3.0 cm and time t = 0.06 sec, is a point on the string moving up or down?  Show work.
(g) (3) What would be the equation y(x,t) if this string were vibrating in the first harmonic ?
(h) (2) Sketch the standing wave pattern between x = 0 and x = L, inclusive, for first harmonic.
 

 

INTERFERENCE:  PHYSICS 5  SP '12 REFERENCE PROBLEMS: QUIZ 3, CH. 16, #33, 35, 70 at www..masteringphysics.com; hints at www.nvaphysics.com.

4C TEST 1 AU’98

15. (20  points)

 



In the diagram above, two speakers are shown. The speakers are driven by a common oscillator at 800 Hz. A point P is  on a line joining the two speakers. As indicated, d1 is the distance of the point from speaker 1 and d2 is the distance from speaker 2. The distance between the two speakers is d1 + d2 = 1.25 m. Assume the speed of sound is
v = 343 m/s.  This configuration is identical to a problem assigned in Quiz 3, Ch 16, # 35 ( see also #33, #70 for comparisons) .  In that exercise you are asked to find  points along the line joining the two speakers where relative minima would be expected. The following question is different.

What are the possible values of d1 so that relative maxima would be expected ?

 

 

4C TEST 2 SP ‘00

16. (30  points)

In the diagram above, two speakers are shown. A common oscillator drives the speakers at  a common frequency   f  .  The two speakers are placed  distances d1 and d2 from a detector D.

The speakers are first  arranged as shown in Fig. A.   In Fig. A,  the two distances d1 and d2 are equal.  Speaker 2 is then moved to the left. As Speaker 2 is moved, the sound decreases once to a minimum sound level of zero, then increases once again to a maximum at the final distance d2 = 1.518 m shown in Fig. B. Assume the speed of sound is  v = 343 m/s.  Assume d1 = 1.000 m.

 

(a)   (12 points) What is the wavelength  of  the sound waves emitted from the speakers?

(b)   (6 points)   What is the frequency f  ?

(c)    (12 points)  Suppose that Speaker 2 is moved more to the left as shown in Fig. C until the sound decreases once again to a sound level of zero. What is the distance d2  shown in Fig. C?

 

 

4C TEST 2  SP07

17. (40) Two identical speakers are located at vertically displaced points A and B.  The speakers are a distance d apart. These two  loud speakers are driven by the same amplifier.  They produce sound waves with a frequency of 784 Hz. Take the speed  of sound in air to be 344 m/s . A small microphone is moved out from point B along the horizontal x-axis perpendicular to the vertical line connecting A and B. See diagram.

(a) (30 )What is the smallest non-zero value of d for which constructive interference occurs at x = 4.00 m?

(b) (10) What is the length  h of the hypotenuse formed by the right triangle shown below? (Hint: You might have already discovered usefulness of the Pythagorean Theorem in this problem.)