Special Relativity: Kinematics
Problems on Lorentz Transformations and Minkowski Diagrams
Problem : Show that the if we have tanhθ = v/c , the Lorentz transformations can be written as:
|Δx = Δx'coshθ + cΔt'sinhθ|
|Δct = Δx'sinhθ + cΔt'coshθ|
Problem : The Lorentz transformation expressed in the problem above may be represented in matrix form by:
Show that is you apply one Lorentz Transformation with tanhθ 1 = v 1/c followed by another Lorentz transformation with tanhθ 2 = v 2/c , the result is also a Lorentz transformation with tanh(θ 1 + θ 2) = v .
Problem : In the reference frame of an outside observer two particles move towards each other, both with velocity v . The angle between them is 2θ as shown in the figure below. What is the speed of one of the particles as viewed by the other?
Problem : Show that the angle between the x and x' axes, as shown in , on a Minkowski diagram is given by tanθ 2 = v/c . Also, determine the size of one unit on the x' axis.
Problem : Use a Minkowski diagram to solve the following problem. Frame F' moves at a speed v with respect to frame F along the x -direction. A 1-meter stick (as measured in F' ) lies along the x' axis, at rest in F' . An observer in F measures the length of the stick. What is the result?
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