CH 2, Sample Problem 2

Transcription

1 Special Relativity

2 CH 2, Sample Problem 2 A river flows at uniform speed v w = 1.0 m/s between parallel shores a distance D = 120 m apart. A kayaker can paddle at 2.0 m/s relative to the water. a) If the kayaker starts on one shore and always paddles perpendicular to the shoreline, how long does it take him to reach the opposite shore, and how far downstream is he swept? b) If instead he wants to reach a point on the opposite shore straight across from the starting point, at what angle should he paddle relative to the stream flow direction so as to arrive in the shortest time? What is this shortest time?

3 CH 4, Sample Problem 2 Clock A is at rest in our frame of reference and Clock B is moving at speed (3/5)c relative to us. Just as Clock B passes Clock A, both read 12 am. a) When Clock A reads 5 am, what does Clock B read, as observed in our frame? b) When Clock B reads the time found in a), what does Clock A read as observed in Clock B s frame?

4 Quiz Question 1 Alpha Centauri, our nearest neighbor, is at a distance of 4 light years from us. Suppose travelers move at a speed of (4/5)c relative to Earth. According to clocks on Earth and Alpha Centauri, how long does it take the spaceship to arrive? 1. 3 years 2. 4 years 3. 5 years /3 years

5 Quiz Question 2 Alpha Centauri, our nearest neighbor, is at a distance of 4 light years from us. Suppose travelers move at a speed of (4/5)c relative to Earth. According to clocks on the spaceship, how long does the trip take? 1. 3 years 2. 4 years 3. 5 years /3 years

6 CH 4, Sample Problem 3 A particular muon lives for a time of 2.0 µs in its rest frame. If it moves in the laboratory a distance d = 800 m from creation to decay, how fast did it move according to the lab frame?

7 CH 5, Sample Problem 1 Barnard s star is 6.0 c-yrs from the Sun in the rest frame of the Sun and star. A spaceship departs the Sun at speed (4/5)c bound for the star, when both the Sun clock and spaceship clock read t=t =0. In the rest frame of the spaceship a) How far apart are the Sun and star? b) Draw two pictures in the spaceship frame, one when the ship is beside the Sun, and one when the star has reached the ship. c) What will the ship clock read when the star arrives at the ship?

8 Quiz Question 3 Alpha Centauri, our nearest neighbor, is at a distance of 4 c-yrs from us. Suppose travelers move at a speed of (4/5)c relative to Earth. According to the spaceship, what is the distance to the star? c-yrs 2. 4 c-yrs /3 c-yrs 4. None of the above

9 Three rules: p 1. Moving clocks run slow by the factor 1 v2 /c p Moving objects are contracted by the factor 1 v2 /c Two clocks synchronized in their own frame are NOT synchronized in other frames. The front (leading) clock reads an earlier time (lags) the chasing clock by t 0 = vd/c 2 Notice: p p Summary where D is the rest distance between them. In rule 1, an observer with a clock finds that pa moving clock runs at a different rate by the multiplicative factor 1 v2 /c 2. In rule 3, an observer finds that two moving clocks (mutually at rest), run at the same rate as one another, however, the observer has to add t 0 to the reading of the leading clock to get the reading of the chasing clock.

10 CH 6, Sample Problem 1 A spaceship of rest length D = 500 m moves by us at speed v = (4/5)c. There are two clocks on the ship, at its nose and tail, synchronized with one another in the ship s frame. We on the ground have three clocks A, B, and C, spaced at 300 m intervals, and synchronized with one another in our frame. Just as the nose of the ship reaches our clock B, all three of our clocks as well as clock N in the nose of the ship read t = 0. a) At this time t = 0 (to us) what does the clock in the ship s tail read? b) How long does it take the ship s tail to reach us at B? c) At this time, when the tail of the ship has reached us, what do the clocks in the nose and tail read?

11 CH 6, Sample Problem 2 Sketch the spaceship of Sample Problem 1 in the ship s frame a) when clock B passes clock N, and b) when clock B passes clock T. In each sketch label the readings of all five clocks, A, B, C, N, and T.

12 CH 6, Sample Problem 3 Explorers board a spaceship and proceed away from the Sun at (3/5)c. Their clocks read t=0, in agreement with the clocks of earthly observers, at the start of the journey. When the explorers clocks read 40 years, they receive a light message from Earth indicating that the government has fallen. a) Draw a set of three pictures in the Sun s frame, one for each of the three important events in the story. b) Draw a set of three pictures in the ship s frame, one for each of the three important events. c) In the explorers frame, how far are they from the Sun when they receive the signal? d) According to observers in the explorers frame, what time was the message sent? e) At what time do the stay-at-homes say the message was sent? f) According to the explorers, how far from the Sun were they when the message was sent? g) According to the stay-at-homes, how far away was the spaceship when the message was sent?

13 CH 8, Sample Problems 2 & 3 A bus of rest length 15.0 m is barreling along the interstate V=(4/5)c. The driver in the front and a passenger in the rear have synchronized their watches. Parked along the road are several Highway Patrol cars, with synchronized clocks in their mutual rest frame. Just as the rear of the bus passes one of the patrol cars, both the clock in the patrol car and the passenger s watch read exactly 12:00 noon. a) When its clock reads 12:00 noon, a second patrol car happens to be adjacent to the front of the bus. How far is the second patrol car from the first? b) What does the bus driver s watch read according to the second patrol car? The passenger who had been sitting at the back of the speeding bus in SP 2 starts to run forward at speed (3/5)c relative to the bus. How fast is the passenger moving as measured by the parked patrol cars?

14 CH 9, Sample Problem 1 Figure 9.1 graphs how two different coordinate systems (x, y and x, y ), one rotated relative to the other, represent the position of a point in ordinary 2D space. The figure helps us visualize the transformation Eqs Find an analogous graph for the Lorentz transformation between coordinates t, x and t, x, as given by Eqs. 9.4.

15 CH 9, Sample Problem 2 A spacetime shoot-out. Spaceships B and C, starting at the same location when each of their clocks reads zero, depart from one another with relative velocity (3/5)c. One week later according to B s clocks, B s captain goes berserk and fires a photon torpedo at C. Similarly, when clocks on C read one week, C s captain goes crazy and fires a photon torpedo at B. Draw 2D spacetime diagrams of events in B s frame Which ship gets hit first?

16 CH 10, Sample Problem 1 A particle of mass m moves with speed (3/5)c. Find its momentum if it moves (a) (b) purely in the x direction, at an angle of 45 to the x and y axes.

17 CH 10, Sample Problem 2 A particle at rest decays into a particle of mass m moving at (12/13)c and a particle of mass M moving at (5/13)c. Find m/m, the ratio of their masses.

18 CH 10, Sample Problem 3 Starship HMS Pinafore detects an alien ship approaching at (3/5)c. The aliens have just launched a 1-tonne (1000 kg) torpedo toward the Pinafore, moving at (3/5)c relative to the alien ship. The Pinafore immediately erects a shield that can stop a projectile if and only if the magnitude of the projectile s momentum is less than 6.0 x kg m/s. (a) (b) (c) (d) Find the torpedo s momentum four-vector in the alien s rest frame. Find the torpedo s momentum four-vector in Pinafore s frame. Does the shield stop the torpedo? Find the torpedo s velocity relative to the Pinafore.

19 CH 11, Sample Problem 3 A photon of energy 12.0 TeV (1TeV = ev) strikes a particle of mass M 0 at rest. After the collision there is only a single final particle of mass M, moving at speed (12/13)c. Using ev units, find a) the momentum of the final particle, b) the mass M, c) the mass M 0.

20 CH 11, Sample Problem 4 Spaceship A fires a beam of protons in the forward direction with velocity v= (4/5)c at an alien ship B fleeing directly away at velocity V=(3/5)c. Transforming the beam s energy-momentum four-vector using the Lorentz transformation of Eqs , find the beam s velocity v in the frame of ship B.