Phys 2D soln1 - of the astronaut as measured by Earth can...

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1a Faraday’s force law does not satisfy the principle that the laws of mechanics are the same in all inertial reference frames because it explicit depends on the velocity of the particle. 1b A Lorentz transformation relates the space and time coordinates of two inertial observers moving with a relative speed v . Postulates: 1, the laws of physics are the same in every inertial reference frame, ( x, t ) ( x , t ), v ↔ - v . 2, the speed of light is the same in all inertial reference frames, if u x = c then u x = c . 1c If F = ma were always true then the speed of a particle of finite mass subject to a constant force would have constant acceleration and eventually exceed the speed of light, which is not possible according to Einstein’s theory of special relativity A particle of finite mass would accelerate less and less as it approached the speed of light. 1d If the distance to the star is measured from Earth then it is possible because the lifetime
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Unformatted text preview: of the astronaut as measured by Earth can increase arbitrarily due to time dilation. 2a Since E = mc 2 / ± 1-v 2 c 2 ² 1 / 2 , if E is to remain finite as m → 0, then v must approach c . Thus, a massless particle must travel at the speed of light. E | p | = c 2 | u | | u | = c E = pc 2b K ≡ E-E K = ( γ-1) mc 2 γ = ³ 1-u 2 c 2 ´-1 / 2 γ ≈ 1 + 1 2 u 2 c 2 + ··· K ≈ ³ 1 + 1 2 u 2 c 2-1 ´ mc 2 K = 1 2 mu 2 2c 1 dx = γ ( dx-vdt ) dt = γ ± dt-v c 2 dx ² u x ≡ dx dt u x ≡ dx dt u x = u x-v 1-u x v c 2 if v ± c this reduces to u x = u x-v 3 Use conservation of energy. The energy of the initial particle is Mc 2 . The energy of each of the 2 photons is pc From conservation of momentum, the photons must be traveling in opposite directions. Mc 2 = 2 pc p = 1 2 Mc 2...
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