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Question 1 [14 marks] Let S and S' be inertial frames. Frame S' moves at velocity v with respect to frame S, in the common (positive) x-direction. Measurements of events in the two frames, denoted respectively by (x, y, z, t) and (x', y', z', t'), are related by the Lorentz transformation x' = ?(x - vt), y' = y, z' = z, t' = ?(t - rac{v}{c^2}x), where ? = rac{1}{sqrt{1-frac{v^2}{c^2}}} and c is the speed of light. (a) Show that the quantity (x'^2 - c^2t'^2) is invariant under this Lorentz transformation. [6 marks] (b) By algebraically solving for x and t, find the inverse Lorentz transformation, i.e., express (x, y, z, t) in terms of (x', y', z', t'). [6 marks] (c) Explain why you could have expected this result, based on the velocity of frame S relative to frame S'. [2 marks] Question 2 [12 marks] Consider inertial frames S and S'. Frame S' moves at velocity v with respect to frame S, in the common (positive) x-direction. Measurements of an event in two frames (x, y, z, t) and (x', y', z', t') are related by the Lorentz transformation: x' = ?(x - vt), y' = y, z' = z, t' = ?(t - rac{v}{c^2}x), where ? = rac{1}{sqrt{1-frac{v^2}{c^2}}} and c is the speed of light. (a) A photon has velocity u' = (0, 0, c) relative to frame S' where u' = (u'_x, u'_y, u'_z). Find the velocity, u, of the photon relative to frame S. [8 marks] (b) Explain how your result is consistent with the constancy of the speed of light. [4 marks]

          Question 1 [14 marks]
Let S and S' be inertial frames. Frame S' moves at velocity v with respect to frame S, in the common (positive) x-direction. Measurements of events in the two frames, denoted respectively by (x, y, z, t) and (x', y', z', t'), are related by the Lorentz transformation
x' = ?(x - vt), y' = y, z' = z, t' = ?(t - rac{v}{c^2}x),
where ? = rac{1}{sqrt{1-frac{v^2}{c^2}}} and c is the speed of light.
(a) Show that the quantity
(x'^2 - c^2t'^2)
is invariant under this Lorentz transformation.
[6 marks]
(b) By algebraically solving for x and t, find the inverse Lorentz transformation, i.e., express (x, y, z, t) in terms of (x', y', z', t').
[6 marks]
(c) Explain why you could have expected this result, based on the velocity of frame S relative to frame S'.
[2 marks]
Question 2 [12 marks]
Consider inertial frames S and S'. Frame S' moves at velocity v with respect to frame S, in the common (positive) x-direction. Measurements of an event in two frames (x, y, z, t) and (x', y', z', t') are related by the Lorentz transformation:
x' = ?(x - vt), y' = y, z' = z, t' = ?(t - rac{v}{c^2}x),
where ? = rac{1}{sqrt{1-frac{v^2}{c^2}}} and c is the speed of light.
(a) A photon has velocity
u' = (0, 0, c)
relative to frame S' where u' = (u'_x, u'_y, u'_z). Find the velocity, u, of the photon relative to frame S.
[8 marks]
(b) Explain how your result is consistent with the constancy of the speed of light.
[4 marks]

$Question 1 [14 marks] Let S and S' be inertial frames. Frame S' moves at velocity v with respect to frame S, in the common (positive) x-direction. Measurements of events in the two frames, denoted respectively by (x, y, z, t) and (x', y', z', t'), are related by the Lorentz transformation x' = ?(x - vt), y' = y, z' = z, t' = ?(t - racvc^2x), where ? = rac1sqrt1-fracv^2c^2 and c is the speed of light. (a) Show that the quantity (x'^2 - c^2t'^2) is invariant under this Lorentz transformation. [6 marks] (b) By algebraically solving for x and t, find the inverse Lorentz transformation, i.e., express (x, y, z, t) in terms of (x', y', z', t'). [6 marks] (c) Explain why you could have expected this result, based on the velocity of frame S relative to frame S'. [2 marks] Question 2 [12 marks] Consider inertial frames S and S'. Frame S' moves at velocity v with respect to frame S, in the common (positive) x-direction. Measurements of an event in two frames (x, y, z, t) and (x', y', z', t') are related by the Lorentz transformation: x' = ?(x - vt), y' = y, z' = z, t' = ?(t - racvc^2x), where ? = rac1sqrt1-fracv^2c^2 and c is the speed of light. (a) A photon has velocity u' = (0, 0, c) relative to frame S' where u' = (u'x, u'y, u'z). Find the velocity, u, of the photon relative to frame S. [8 marks] (b) Explain how your result is consistent with the constancy of the speed of light. [4 marks]$

Added by Rocio M.

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