Problems on Lorentz Transformations and Minkowski Diagrams
Problem 3.1:
Show that the if we have
tanhθ = v/c, the Lorentz transformations can be written as:
| Δx = Δx'coshθ + cΔt'sinhθ |
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| Δct = Δx'sinhθ + cΔt'coshθ |
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[Solution]
Problem 3.2:
The Lorentz transformation expressed in the problem above may be represented in matrix form by:
Show that is you apply one Lorentz Transformation with
tanhθ1 = v1/c followed by another
Lorentz transformation with
tanhθ2 = v2/c, the result is also a Lorentz transformation with
tanh(θ1 + θ2) = v.
[Solution]
Problem 3.3:
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.
[Solution]
Problem 3.4:
Show that the angle between the
x and
x' axes, as shown in the minkowski diagram3.2, on a
Minkowski diagram is given by
tanθ2 = v/c. Also, determine the size of one unit on the
x' axis.
[Solution]
Problem 3.5:
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?
[Solution]
[Solution]
