On the diagram, sketch the path of the light pulse between ${{\text{M}}_{\text{1}}}$ and ${{\text{M}}_{\text{2}}}$ as observed by Jill.

(i)     State, according to Jill, the distance moved by the spaceship in time $\Delta t$.

(ii)     Using a Galilean transformation, derive an expression for the length of the path of the light between ${{\text{M}}_{\text{2}}}$ and ${{\text{M}}_{\text{1}}}$.

State, according to special relativity, the length of the path of the light between ${{\text{M}}_{\text{1}}}$ and ${{\text{M}}_{\text{1}}}$ as measured by Jill in terms of $c$ and $\Delta t$.

The time for the pulse to travel from ${{\text{M}}_{\text{2}}}$ to ${{\text{M}}_{\text{1}}}$ as measured by Ben is $\Delta t'$. Use your answer to (b)(i) and (c) to derive a relationship between $\Delta t$ and $\Delta t'$.

According to a clock at rest with respect to Jill, a clock in the spaceship runs slow by a factor of 2.3. Show that the speed $v$ of the spaceship is 0.90c.

any diagonal line as shown;

(i)     $v\Delta t$;

(ii)     speed of pulse ${({c^2} + {v^2})^{\frac{1}{2}}}$;

distance $= {({c^2} + {v^2})^{\frac{1}{2}}}\Delta t$;

Award [2] for bald correct answer.

$c\Delta t$;

$d = c\Delta t'$;

from Pythagoras ${d^2} = {c^2}\Delta {t'^2} = {c^2}\Delta {t^2} - {v^2}\Delta {t^2}$;

$\Delta t = \frac{{\Delta t'}}{{\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} }}$;

recognize that $2.3 = \frac{1}{{\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} }}$;

some evidence of rearranging e.g. $v = \sqrt {\frac{{{{[2.3]}^2} - 1}}{{{{[2.3]}^2}}}}$;

$= 0.90{\text{c}}$

Most candidates were able to draw the correct path of the light.

Parts (b), (c) and (d) effectively dealt with the derivation of the time dilation formula and here many candidates had problems often relying on guesswork and half-remembered proofs rather than follow the logical development of the questions. The calculation was often done well but with the usual confusion between the times.

Parts (b), (c) and (d) effectively dealt with the derivation of the time dilation formula and here many candidates had problems often relying on guesswork and half-remembered proofs rather than follow the logical development of the questions. The calculation was often done well but with the usual confusion between the times.

Parts (b), (c) and (d) effectively dealt with the derivation of the time dilation formula and here many candidates had problems often relying on guesswork and half-remembered proofs rather than follow the logical development of the questions. The calculation was often done well but with the usual confusion between the times.