Homework 3 : Discovery of the expanding universe and the Special Theory of Relativity

(due at the start of class on the 3rd October 2002)

 

 

  1. [5 points] What event in 1885 misled astronomers to think that the Andromeda “nebula” was nearby and situated within the Milky Way?  Why was this event misleading?  What observation allowed the true nature of the Andromeda galaxy to be finally determined?

 

  1. Suppose that we measure the spectrum of a distant galaxy.   We see absorption lines in the spectrum with a wavelength of 484nm (1nm=10-9m) but we know from laboratory measurements that the true intrinsic wavelength of the absorption lines should be 440nm. 
    1. [3 points] Calculate the redshift of the galaxy.
    2. [3 points] Assuming the non-relativistic formula for the redshift-velocity conversion, calculate the apparent velocity of recession for the galaxy.
    3. [4 points] Assuming that Hubble’s constant is 65km/s/Mpc, calculate the distance to the galaxy.

 

  1. [5 points] Gamma-Ray Bursts (GRBs) are extremely violent explosions in space that probably arise from a particularly catastrophic collapse of a massive star.  When a GRB detonates, it sends a powerful blast wave through space that travels at almost the speed of light.  Suppose a particular GRB produces a blast wave with a velocity of v=0.9995c, where c is the speed of light.  What is the Lorentz factor corresponding to this velocity?  Suppose you were in a spacecraft riding along with this wave as it was engulfing a planet with a diameter of R=10,000km.  What shape and size would the planet appear to have (just prior to being engulfed – it would probably be destroyed rapidly once it has been engulf!)?

 

  1. [5 points] Two spaceships (A and B) approach your spacecraft at 9/10 the speed of light (0.9c) from opposite directions.  They send out radio messages.   What is the speed you measure for the radio waves from A and B?  What is the speed of the radio waves from A as measured by B.  What speed does B measure for your motion?  How fast does spaceship B observe spaceship A to be moving?

 

  1. [5 points extra credit – this is fairly hard!] You have just bought a nice new house, but, to your horror, discover that the garage is only half the length of your car!   But, having just learnt about the special theory of relativity, you come up with a cunning plan… Suppose you drove your car into the garage at 90% the speed of light.  Length contraction will then shrink the length of the car by a factor of g=2.3.  So the car should fit!  The instant the back of the car is inside the garage, you would slam the garage door shut and then stop the car.  As long as the garage door and walls are strong enough, the car would then be trapped inside the garage even though it might try to re-expand once it stops.    You persuade a mechanic to make the necessary enhancements to your car’s engine and brakes, and persuade a builder to strengthen the door and walls of your garage. You then attempt this maneuver.  

    Explain why it goes horribly wrong, and why the above logic is flawed.   Be as clear in your explanation as possible.   [Hint – Think about the following two events.  Event A is “the back of the car enters the garage, and the garage door closes”.  Event B is “the front of the car hits the front of the garage”.   Think about the order of Events A and B in the two relevant frames of reference.  Is it possible to stop the car in time without violating some basic laws of physics?]