ps12_soln - Physics 105 Problem Set 12 Solutions Each...

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Unformatted text preview: Physics 105 Problem Set 12 Solutions Each problem is worth 3 points Problem 9.1 An observer sees two spaceships flying apart, each with speed . 99 c relative to the observer. What is the speed of one spaceship as viewed by the other? In order to solve the problem one needs to go to the reference frame related to one of the spaceships and find the velocity of the second one. This is done by the relativistic law of velocity addition v = v 1 + v 2 1 + v 1 v 2 /c 2 ; where v 1 = 0 . 99 c = v 2 . Performing the substitution we get v ' . 99995 c. Problem 9.2 A laser beam is aimed at angle θ above the horizontal axis (the x-axis) as measured in the rest frame of the laser. An observer in a different reference frame sees the laser moving at very high velocity, v , along the x-axis in the observer’s frame. What is the laser beam’s angle above the horizontal in this observer’s frame? (You may leave your answer in terms of β = v/c and the Lorentz factor, γ , as well as v and θ .) Hint: this problem involves two spatial dimensions and one temporal dimension. It might be useful to think in terms of 4-vectors. Consider a pulse of light from the laser. Define two events to be the locations of the laser pulse at different times. Assume that a photon from the laser was emitted at the space-time point ( t = 0 , x = 0 , y = 0) in the (moving) S reference frame. This corresponds to ( t = 0 , x = 0 , y = 0) in the observer’s frame S . Then this photon was registered at the spacetime point t = T , x = cT cos θ , y = cT sin θ . Using Lorentz transformation, we get the spacetime coordinates of this event (registration of the photon) in the S frame as t = γ ( t + vx /c 2 ) = γT (1 + β cos θ ) , x = γ ( x + vt ) = γcT (cos θ + β ) , y = y = cT sin θ ....
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ps12_soln - Physics 105 Problem Set 12 Solutions Each...

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