Question

Let a source emit a beam of light which makes an angle θ' with the x' axis in the S' frame. Imagine that the source is at rest in the S' frame and that the frame moves with velocity v = c β in the x-direction with respect to the S frame. (a) Show that the angle θ which the beam makes with the x-axis in the S frame is obtained from Tan[θ] = 1 - β2 Sin[θ'] Cos[θ'] + β (b) Draw a graph of θ as a function of θ' for two cases (i) γ = 5 / 3 and (ii) γ = 5 where γ = 1 1 - β2 (c) Comment on your interpretation of the graph as to why this is called the headlight effect.

46. Let a source emit a beam of light which makes an angle θ. with the x' axis in the S' frame. Îmagine that the S frame. (a) Show that the angle 0 which the beam makes with the x-axis in the S frame is obtained from Sin e1 (b) Draw agraph ofeasa function of e fortwo cases (i) -5/ 3 and (ii) 1-3 where,. (c) Comment on your interpretation of the graph as to why this is called the headlight effect.

Answer 1

So By velocity addition formula In frame s velocity of right in the x-direction c cos theta = theta variant + c cos theta'/1 + theta variant c ln theta'/c^2 As theta variant = beta c c cos theta = beta c + beta c cos theta'/1 + beta c^2 times cos theta'/c^2 cos theta = (beta + cos theta')/1 + beta cos theta' so By properties of angle of \r\n Height h = squareroot 1 + beta^2 cos theta'^2 + 2 beta cos theta' + beta^2 - cos theta' - 2 beta cos theta' h = squareroot sin theta' tan theta = squareroot 1 - beta^2 sin theta'/beta + cos theta' \r\n r = 5/3 squareroot 1 - beta^2 = 3/5 1 - beta^2 = 9/25 beta^2 = 1 - 9/25 = 16/25 beta = 4/5 tan theta = 3/5 sin theta'/4/5 + cos theta'