IB Physics HL A.5 Galilean and special relativity questions

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Question 1

Paper 2Medium4 marks

In inertial frame $S$ a particle moves along the $x$ -axis as shown in Figure 1. Inertial frame $S'$ moves in the $+x$ -direction at speed $v=0.50c$ relative to $S$ The origins of the frames coincide at $t=t'=0$ Take $c=3.00\times10^8\ \mathrm{m\,s^{-1}}$

The Lorentz transformations are

x'=\gamma(x-vt),\qquad t'=\gamma\left(t-\frac{vx}{c^2}\right), \qquad \gamma=\frac{1}{\sqrt{1-v^2/c^2}}.

(a)

Determine $u/c$ for the particle in frame $S$

[1]

(b)

For two events on the particle’s motion, use the Lorentz transformations to show that

u'=\frac{u-v}{1-uv/c^2}.

[2]

(c)

Use your answers to (a) and (b) to determine $u'/c$ to 2 significant figures.

[1]

Question 2

Paper 2Medium4 marks

A trolley moves at constant velocity in the $+x$ direction relative to the ground. Let $S$ be the ground frame and $S'$ the frame of the trolley. The Galilean transformations are $x'=x-vt$ and $t'=t$ where $v$ is the velocity of $S'$ relative to $S$

(a)

Using the transformations for the positions of an object at two different times, show that its velocity $u'$ measured in $S'$ is related to its velocity $u$ measured in $S$ by $u'=u-v$

[2]

(b)

A puck moves at $+2.5\ \mathrm{m\,s^{-1}}$ relative to the ground while the trolley moves at $+4.0\ \mathrm{m\,s^{-1}}$ relative to the ground. Determine the puck’s velocity relative to the trolley, including its direction.

[2]

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