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					                                    Relativity
Multiple Choice
1.   A 1000-kg automobile moving with a speed of 24 m/s relative to the road
     collides with a 500-kg automobile initially at rest. If the two stick together, what
     is the velocity in m/s of the two cars after the collision according to an observer
     in a truck moving 10 m/s in the same direction as the moving cars?
     a.   6
     b.   11
     c.   24
     d.   26
     e.   31

2.   A 1000-kg automobile moving with a speed of 24 m/s collides with a 500-kg car
     initially at rest. If the two stick together, what is the velocity (in m/s) of the two
     cars after the collision relative to an automobile moving in the same direction at
     15 m/s?
     a.   14
     b.   16
     c.   24
     d.   48
     e.   1.0

3.   A boat has an initial velocity of 2 m/s in the y-direction on a stream which is
     moving in the x-direction at 1 m/s. The boat is accelerating in its direction of
     motion at 1 m/s2. What is the acceleration of the boat (in m/s2) relative to the
     water?
     a.   1i
     b.   2i
     c.   3i + j
     d.   3i – j
     e.   1j

4.   A spaceship moves at a speed of 0.95 c away from the Earth. It shoots a star wars
     torpedo toward the Earth at a speed of 0.90 c relative to the ship. What is the
     velocity of the torpedo relative to the Earth? (The direction in which the
     spaceship moves is the positive direction.)
     a.   –0.35 c
     b.   0.35 c
     c.   0.06 c
     d.   –0.06 c
     e.   0




                                           309
5.   A satellite moves east, taken as the positive x-axis direction, at a speed of 0.5 c
     and a spaceship moves toward it (to the west) at a speed of 0.8 c as measured by
     an observer on the Earth. The speed of the satellite measured by an observer in
     the spaceship is
     a.   8.7 c
     b.   0.21 c
     c.   –0.21 c
     d.   0.93 c
     e.   –0.93 c

6.   Boat 1 goes directly across a stream a distance L and back taking a time t1. Boat 2
     goes down stream a distance L and back taking a time t2. If both boats had the
     same speed relative to the water, which of the following statements is true?
     a.   t2 > t1
     b.   t2 < t1
     c.   t2 = t1
     d.   t2 = 2t1
     e.   t2 = 0.5t1

7.   A fancy sports car passes Big Ben at a speed of 0.6 c. What time interval will the
     driver measure for a one-second interval on the large clock?
     a.   1.67 s
     b.   0.8 s
     c.   1.25 s
     d.   0.6 s
     e.   1.0 s

8.   A fancy sports car moves past an observer on a corner at a speed of 0.6 c. When
     the observer indicates a one-second interval has passed, what time interval will
     be shown on the driver’s watch?
     a.   1.67 s
     b.   0.8 s
     c.   1.25 s
     d.   0.6 s
     e.   1.0 s
                                                                          Relativity    311



9.    Two fireworks explode at the same position on the 4th of July. A stationary
      observer notices that the time interval between the two events was 5.00 seconds.
      A second observer flies past the fireworks at a speed of 0.600 c. What value does
      she obtain when she measures the time interval between the two explosions?
      a.   8.33 s
      b.   6.25 s
      c.   4.0 s
      d.   3.0 s
      e.   5.0 s

10.   The half-life of a muon is 2.2 s as measured in a stationary reference frame.
      What is the half life of the muon (in s) when it is moving with a speed of
      v = 0.800 c?
      a.   8.13
      b.   2.75
      c.   3.67
      d.   15.8
      e.   1.32

11.   The half-life of a muon is 2.2 s. How fast is it moving relative to an observer
      who says its half-life is 4.4 s?
      a.   0.87 c
      b.   0.75 c
      c.   0.97 c
      d.   0.72 c
      e.   0.50 c

12.   A spaceship moving past the Earth with a speed of 0.800 c signals to the Earth
      with pulsed laser photons emitted at 10 second intervals according to the
      spaceship’s clock. According to observers on Earth who see the flashes, the time
      interval they measure is
      a.   13.4 s
      b.   16.7 s
      c.   12.5 s
      d.   9.7 s
      e.   6.0 s
13.   A 30-year-old woman takes a trip on a rocket, leaving her 20-year-old brother
      behind. She travels at a speed of 0.8 c, and is gone 20 years, according to the
      younger brother. When she returns, how many years older/younger is she than
      her brother?
      a.   2 years younger
      b.   2 years older
      c.   3 years older
      d.   10 years older
      e.   8 years older

14.   A jet plane travels around the world at 2000 mi/hr (894 m/s). Two accurate
      atomic clocks measure the times of flight, one on board the plane and the second
      on Earth. If it takes 12 hours to complete the journey, what will the time
      difference (in s) be?
      a.   0.75
      b.   0.50
      c.   0.25
      d.   1.0
      e.   0.19

15.   A meterstick is shot from a meterstick projector at a speed of 0.90 c. How long
      will it be relative to an observer’s frame of reference?
      a.   2.3 m
      b.   0.91 m
      c.   1.0 m
      d.   0.44 m
      e.   0.81 m

16.   A starship navigator measures the distance between the Earth and the sun. If the
      ship is moving at a speed of 0.90 c, instead of obtaining 93 million miles, the
      navigator measures a distance (in millions of miles) of
      a.   41.
      b.   30.
      c.   80.
      d.   215.
      e.   90.

17.   An astronaut traveling with a speed v = 0.9 c holds a meterstick in his hand. If he
      measures its length, he will obtain a value of
      a.   1m
      b.   2.3 m
      c.   0.19 m
      d.   0.43 m
      e.   0.81 m
                                                                             Relativity   313



18.   An electron (m = 9.11 × 10–31 kg) has a speed of 0.50 c. Determine the difference
      between its relativistic kinetic energy and the kinetic energy calculated without
      considering relativity.
      a.   3.0 × 10–15 J
      b.   2.0 × 10–15 J
      c.   1.5 × 10–15 J
      d.   2.4 × 10–15 J
      e.   1.8 × 10–15 J

19.   An electron has a kinetic energy that is twice its rest energy. Determine its speed.
      a.   0.76 c
      b.   0.81 c
      c.   0.94 c
      d.   0.54 c
      e.   0.87 c

20.   An electron (m = 9.1 × 10–31 kg) has a speed of 0.90 c. What is the difference
      between its relativistic momentum and its non-relativistic momentum (in kg
      m/s)?
      a.   4.3 × 10–22
      b.   3.2 × 10–22
      c.   5.4 × 10–22
      d.   6.5 × 10–22
      e.   2.5 × 10–22

21.   A proton’s rest mass is 1.67 × 10–27 kg. Calculate its total energy when it is
      accelerated to a speed of 0.80 c.
      a.   1.5 × 10–10 J
      b.   1.0 × 10–10 J
      c.   2.5 × 10–10 J
      d.   2.0 × 10–10 J
      e.   7.5 × 10–10 J

22.   A proton’s rest mass is 1.67 × 10–27 kg. Calculate its kinetic energy when it is
      accelerated to a speed of 0.80 c.
      a.   1.0 × 10–10 J
      b.   1.5 × 10–10 J
      c.   –2.0 × 10–10 J
      d.   2.5 × 10–10 J
      e.   7.5 × 10–10 J
23.   Energy is released during a nuclear reaction due to a conversion between mass
      and energy. Mass is not conserved. The initial and final amounts are different. If
      a total of 1 gram of mass is “missing”, how much energy has been released?
      a.   9.0 × 1011 J
      b.   9.0 × 1017 J
      c.   9.0 × 1015 J
      d.   9.0 × 1013 J
      e.   9.0 × 1016 J

24.   Assume a gram of a substance marketed as “Pure Energy” is annihilated by a
      gram of a second substance “Anti-Pure Energy.” How long would the energy
      released power a city which uses 109 watts of power?
      a.   40 hrs
      b.   25 hrs
      c.   50 hrs
      d.   65 hrs
      e.   50 000 hrs

25.   A proton has a total energy of 2.5 × 10–10 J. How fast is it moving?
      (M = 1.67 × 10–27 kg)
      a.   0.9 c
      b.   0.8 c
      c.   0.7 c
      d.   0.6 c
      e.   0.4 c

26.   A spaceship from another galaxy passes over the solar system directly above a
      radial line from the sun to the Earth. (We measure that distance to be
      1.5 × 1011 m.) On Earth, the spaceship is observed to be traveling at a speed of
      0.8c, for which γ = 5/3. As measured on Earth it takes the spaceship 625 seconds
      to travel from the sun to Earth. When a scientist in the spaceship measures the
      Earth-sun distance and the time it takes her to travel that distance, she finds the
      results are respectively
      a.   9 × 1010 m; 375 s.
      b.   9 × 1010 m; 625 s.
      c.   1.5 × 1011 m; 625 s.
      d.   2.5 × 1011 m; 625 s.
      e.   2.5 × 1011 m; 1042 s.
                                                                            Relativity   315



27.   Fireworks go off at the same time according to Earth clocks in two cities, Alum
      and Boron, that are 300 km apart. The people in a spaceship that is flying in a
      straight line from Alum to Boron at 0.8c also observe the fireworks. Do they see
      the fireworks in the two cities simultaneously? If the people in the spaceship say
      the fireworks were not simultaneous in Alum and Boron, how long before or
      after the fireworks flashed at Alum did the fireworks flash at Boron according to
      their calculations? (The spaceship is directly over Alum when the fireworks
      flash.)
      a.   Yes; 0
      b.   Before; 1 × 10–3 s
      c.   After; 1 × 10–3 s
      d.   Before; 1.33 × 10–3 s
      e.   After; 1.33 × 10–3 s

28.   The first intergalactic spaceship is headed toward the Magellanic Clouds at a
      speed of 0.8c. The spaceship is 1000 m long. Clocks at the front and the rear of the
      spaceship both read 3:00 P.M. Can it be 3:00 P.M. simultaneously at the front and
      the back of the spaceship?
                                ∆x
      a.   No, because ∆t = γβ      , where ∆x is the length of the spaceship.
                                 c
      b.   No, because one clock has to move after being synchronized with the other.
      c.   Yes, because ∆x in (a) is zero for different points of the same spaceship.
      d.   Yes, because two clocks at rest relative to each other can be synchronized by
           means of light signals when the distance between them is known.
      e.   The question cannot be answered unless we know the object relative to
           which the spaceship’s velocity is 0.8c.

29.   As a spaceship heads directly to Earth at a velocity of 0.8c, it sends a radio signal
      to Earth. When those radio waves arrive on Earth, their velocity relative to Earth
      is
      a.   0.8 c.
      b.   c.
      c.   1.8 c.
      d.           2
             c2 + vE , where vE is the velocity of the Earth.
      e.      08 2     2
             ( . c) + vE , where vE is the velocity of the Earth.

30.                                                 9979 108 m/s when the
      The speed of FM waves will be observed to be 2.   ×
      antenna emitting the waves is
      a.   at rest relative to the receiving antenna.
      b.   moving to the right of the detecting antenna at 0.5 c.
      c.   moving to the left of the detecting antenna at 0.5 c.
      d.   moving as described in a, b or c above.
      e.   moving at 2.9979 × 108 m/s.
31.   Captain Jirk reports to headquarters that he left the planet Senesca 1. × 104
                                                                            88
      seconds earlier. Headquarters sends back the message: “Was that spaceship
      proper time?” It will be spaceship proper time if it was
      a.   measured by one clock fixed at one spot on Senesca.
      b.   measured by one clock fixed at one spot on the spaceship.
      c.   measured by a clock on Senesca at departure and by a clock on the
           spaceship when reporting.
      d.   measured by a clock on the spaceship when departing and by a clock on
           Senesca when reporting.
      e.   calculated by dividing the distance from Senesca according to Senesca by
           the speed of the spaceship.

32.   In a classroom on the first spaceship to an extrasolar planet – there are children
      because the trip will take 200 years – a teacher is showing charge Q uniformly
      distributed along a conducting rod of length L0 to produce linear charge density
      λ0 . As observed on Pluto when the spaceship passes it at 0.80 c, the linear charge
      density λ ′ is
      a.   0. λ0 .
            36
      b.   0. λ0 .
            60
      c.   0. λ0 .
            80
      d.   1. λ0 .
            00
      e.   1. λ0 .
            67

33.   The quantity which does not change in magnitude from that observed in system
      S when observed in system S′ moving away from system S at speed v is
      a.   m a.
      b.   mv.
      c.   ( − 1) 2 .
            γ    mc
      d.   E 2 − p2c2 .
      e.   x2 + y2 + z2 .

34.   Hanna, at rest in her spaceship which is moving past Earth at 0.8 c, observes a
      neutron at rest relative to her spaceship decay into a proton, an electron and a
      neutrino. She notes that the total momentum, P0 , of the decay products is zero
      after the decay. According to an observer on Earth, the magnitude of the total
      momentum of the decay products, P ′ , is
      a.   0.
      b.      6    08
           0. m n ( . c)
      c.   0. m n ( . c).
              8    08
      d.   1. m n ( . c).
              00 0 8
      e.   1. m n ( . c).
              67 0 8
                                                                          Relativity   317



35.   A spaceship leaves Earth and maintains a constant force by means of a nuclear
      engine. As the speed of the spaceship increases, an observer on Earth finds that
      relative to her the magnitude of the spaceship’s acceleration is
      a.   0.
      b.   decreasing.
      c.   constant.
      d.   increasing.
      e.   proportional to the kinetic energy of the spaceship.

36.   The 500 m-long spaceship Springbrake is at rest on the planet Hitest for
      refueling. Another spaceship, Summerbrake, passes parallel to Springbrake at
      0.600c. The crew on Springbrake measure the length of Summerbrake as it passes
      and find that the length they measure is exactly the same as the known 500 m
      rest length of Springbrake. If Summerbrake were at rest next to Springbrake, its
      length would then be measured to be
      a.   180 m.
      b.   320 m.
      c.   500 m.
      d.   625 m.
      e.   781 m.

37.   Two spaceships traveling in opposite directions along parallel lines measure
      their own and the other spaceship’s length while passing one another. The crew
      on spaceship A says that their ship is 1000 m long and that ship B is 800 m long.
      The crew on ship B says that their ship is 1000 m long and that ship A is 800 m
      long. At what speed does each crew say that the other ship is traveling relative to
      their own ship?
      a.   0.36 c
      b.   0.60 c
      c.   0.64 c
      d.   0.80 c
      e.   1.25 c

38.   Clovis wants to lose weight. Clotilde tells him that he will be thin enough for her
      if he travels past her at 0.600 c. If the distance she sees from the front of his
      stomach to his back is then 24.0 cm, what distance in cm does she see when he is
      standing still?
      a.   15.4
      b.   19.2
      c.   24.0
      d.   30.0
      e.   37.5
39.   Captain Ray, on the Galactic Explorer, sets off bright signal lights at the two ends
      of his spaceship so that they are seen as flashing simultaneously by General Kay,
      who is watch the ship go by. General Kay concludes that she was next to the
      midpoint of the ship at the instant when the lights flashed in her reference frame.
      This can only happen if Captain Ray, at the center of the Galactic Explorer,
      a.   sees the lights flash simultaneously in his coordinate system.
      b.   sees the bow (front) light flash before the aft (rear) light.
      c.   sees the aft (rear) light flash before the bow(front light.
      d.   can calculate that the light flashes reach General Kay so that the flash from
           the bow light reaches her before the flash from the aft light.
      e.   can calculate that the light flashes reach General Kay so that the flash from
           the aft light reaches her before the flash from the bow light.

40.   Lydia proposes to send an interstellar probe off in stages. The first stage,
      traveling radially outward at 0.500 c relative to Earth, will send off a second stage
      traveling in the same direction as the first stage at a speed of 0.500 c relative to
      the first stage. Finally, the second stage will send off a third stage traveling in the
      same direction at 0.500 c relative to the second stage. Relative to Earth, the speed
      of the third stage will have a magnitude of
      a.   0.500 c.
      b.   0,800 c.
      c.   0.929 c.
      d.   0.972 c.
      e.   1.500 c.
                                                                           Relativity    319




Open-Ended Problems
41.   In a color television tube, electrons are accelerated through a potential difference
      of 20 000 volts. With what velocity do the electrons strike the screen?

42.   Find the momentum and speed of a proton whose kinetic energy equals its rest
      energy. (The mass of a proton is 938 MeV/c2).

43.   The period of a pendulum is 2.0 s in a stationary inertial frame of reference. What
      is its period when measured by an observer moving at a speed of 0.6 c with
      respect to the inertial frame of reference?

44.   When 1.0 gram of hydrogen combines with 8.0 grams of oxygen, 9.0 grams of
      water is formed. But is this true? During the reaction 2.86 × 105 J of energy is
      released. How much mass is actually lost in this reaction?

45.   A supertrain (rest-length = 100 m) travels at a speed of 0.95 c as it passes through
      a tunnel (rest-length 50 m). As seen by a trackside observer, is the train ever
      completely within the tunnel? If so, by how much?
          Relativity

1.    a          24.   c
2.    e          25.   b
3.    e          26.   a
4.    b          27.   d
5.    d          28.   d
6.    a          29.   b
7.    c          30.   d
8.    c          31.   b
9.    b          32.   e
10.   c          33.   d
11.   a          34.   e
12.   b          35.   b
13.   b          36.   d
14.   e          37.   b
15.   d          38.   d
16.   a          39.   b
17.   a          40.   c
18.   d          41.   v = 0.27c
19.   c          42.   1625 MeV/c, 0.866c
20.   b          43.   2.5 s
21.   c          44.   3.18 × 10–12 kg
22.   a          45.   yes, by 18.8 meters
23.   d