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Discussion Questions
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6.11 Discussion Questions
- (Challenging problem) Relativistic velocity addition. Consider two inertial observers, S and S', whose frames of reference are related by Lorentz transformation (Eq LT (1) and LT (2). Observer S' sees an object moving in the x-direction with speed u' = \dfrac{dx'}{dt'}=\alpha c where \alpha<1. Find u = \dfrac{dx}{dt}, the speed at which Observer S sees the object is moving at, in terms of u'.
Suppose Observer S' sees a photon travelling with speed u'=c, use the relation you have derived to determine the speed of the photon as seen by Observer S. Verify that the second postulate of special relativity is satisfied.
Note: This relation is the relativistic version of the classical velocity addition formula Eq GR (3)].
- Worldline trajectories. Consider a (1+1 D) Minkowski space-time diagram where the vertical axis is c multiply by time, and the horizontal axis is spatial position. A person located at the origin that is not moving and standing perfectly still will have his worldline indicated by the vertical thick black line on the space-time diagram as shown below.
Plot the worldline of the someone on the above spacetime graph if he/she is
- Moving with a constant velocity v such that {\displaystyle \frac{v}{c}=\frac{dx}{d(ct)}<1}
- Move with a constant velocity v=c.
- Move with a constant velocity v>c.
- Begins motion at origin with v=0, and reaches a velocity of v=0.999\,c after some time.
- The barn and pole paradox. Once upon a time, there was a farmer who had a pole too long to store in his barn. After learning about relativity, he instructed his son to run with the pole as fast as he could, such that the moving pole would Lorentz-contract to a size the barn could accomodate. However, his son, who also knew about relativity, argued that the barn should be the one that Lorentz-contract, not the pole. So the fit would be even worse. Who was right? Would the pole fit inside the barn, or would it not?
- Life-time of a muon. A muon is an elementary particle similar to an electron, with negative one electronic charge and a rest mass of 105.7 MeV/c^{2}. Muons are unstable and decay at an exponential rate into electrons (and neutrinos) with a half-life of 1.5 $\mu$s. Muons are produced at the upper atmosphere of Earth by collisions of cosmic rays (energetic protons from outer space) with the atmospheric molecules. Assume that muons are produced at 15 km above Earth surface at a rate of 10^{6} per minute and travel radially down to Earth at a speed of 0.99$c$.
(a) Without relativistic consideration, determine the rate of detection of muons on the Earth surface.
Now with relativistic consideration. The muons see the surface of Earth moving at 0.99$c$. From the muon point of view, the length between the upper atmosphere and the surface of the Earth is not 15 km but rather a "contracted" length.
(b) Find the length that the muons move through (from the muons' perspective).
(c) Using this length, calculate the rate of detection of muons on the Earth surface.
- Re-look at the muon problem but from the perspective of a observer on Earth seeing the muon move. The half-life of the muon can be taken as the "biological clock" of the muon. What is the half-life of the muon as measured by the Earthling?