# 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?

Two particle approaching each other at an angle 2θ .

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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