Monday, January 17, 2011

Mastering Physics: Postulates of Special Relativity

Einstein's theory of special relativity is based on two postulates:
  1. The principle of relativity: The laws of physics are the same in any inertial coordinate system. For example, by watching the action of the balls on a pool table that is on a ship, you cannot tell whether the ship is at the dock or moving through the water at a constant speed. (You can think of an inertial coordinate system as a nonaccelerating coordinate system. There are actually other subtle conditions, but for now they are not of concern.)
  2. The speed of light in vacuum is constant. This says that observers in any inertial coordinate system will measure the same value for the speed of light, independent of the origin of that light.
The simplicity of these assumptions belies their brilliance--they directly contradict our intuitive ideas of relativity, yet by accepting them we can easily build a theory of relative motion that is in accord with all observation.
We now discuss these ideas more fully, showing where they depart from the previously held ideas about relativity.

Part A
Consider a pool game being played on a pool table on the deck of an aircraft carrier near the bow (front). Assume that the carrier is moving north at 25.0 {\rm m/s}. As a result of the initial break, a ball flies over the edge of the table and over the edge of the deck with a horizontal component of velocity directed toward the bow of the ship of 5.00 {\rm m/s}. What is v_{\rm bw}, the horizontal component of the speed with which this ball strikes the water in front of the ship (i.e., what is the speed of the ball relative to the water)?
Hint A.1
Galilean relativity for velocity addition
Express your answer numerically in meters per second. Use ordinary (Galilean) physics.
ANSWER:

  v_{\rm bw}  = 30
  \rm m/s
Part B
You have used the ideas of Galilean relativity--that time and distance are absolute and independent quantities and that velocities therefore add. In particular, the formula for relative velocity addition is v_{\rm bw} = v_{\rm bc} + v_{\rm cw}, where v_{\rm bw} is the velocity of the ball relative to the water, v_bc is the velocity of the ball relative to the carrier, and v_{\rm cw} is the velocity of the carrier relative to the water. These ideas are embodied in Isaac Newton's Principia.
Newton started this seminal work by stating which of the following?
ANSWER:


 
Part C
Which of these factors was definitely not a consideration in Einstein's making the postulate that the speed of light is constant for all observers?
ANSWER:



The exact value of the speed of light was immaterial to Einstein except that it had to be a speed beyond practical comprehension. Also, Michelson measured the speed of light after Einstein's work, which was published in 1905.
 
Part D
We now consider the motion of a flash of light that is emitted from the ship described in Part A. Imagine that there is a flashlight on the pool table that emits a flash of light directed toward the bow of the ship, which is still traveling northward at 25.0 {\rm m/s}. The light encounters two photodiodes on the table that are spaced exactly 1.00 \rm m apart along a north-south axis. What is the time t_table that elapses between when the flash of light encounters the first photodiode and when it strikes the second?

Express your answer in seconds to nine significant figures. Remember that an answer such as 1/300000025 will be accepted at its calculated value (if you do not put commas in your answer). Also, the speed of light in vacuum is c=299,792,458\;\rm m/s.
ANSWER:

  t_table  = 3.3356410×10−9
  \rm s
Part E
Now imagine that there is a similar set of photodiodes, also spaced 1.00 \rm m apart along a north-south axis, mounted on a dock on shore directly ahead of the aircraft carrier. The light from the flashlight on the pool table encounters these two photodiodes. What is the time t_dock that elapses between when the flash of light encounters the first photodiode and when it strikes the second?

Express your answer in seconds to nine significant figures. Remember that an answer such as 1/300000025 will be accepted at its calculated value (if you do not put commas in your answer). Use c = 299,792,458\; {\rm m/s} for the speed of light in vacuum.
ANSWER:

  t_dock  = 3.3356410×10−9
  \rm s
 
Part F
You have now correctly used the key ideas of special relativity. As a consequence of these ideas, what can you conclude?
ANSWER:

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