Two space stations X and Y are at rest relative to each other. The separation of X and Y as measured in their frame of reference is 1.80×10¹¹m.

State what is meant by a frame of reference.

A radio signal is sent to both space stations in (a) from a point midway between them. On receipt of the signal a clock in X and a clock in Y are each set to read zero. A spaceship S travels between X and Y at a speed of 0.750c as measured by X and Y. In the frame of reference of S, station X passes S at the instant that X’s clock is set to zero. A clock in S is also set to zero at this instant.

(i) Calculate the time interval, as measured by the clock in X, that it takes S to travel from X to Y.

(ii) Calculate the time interval, as measured by the clock in S, that it takes S to travel from X to Y.

(iii) Explain whether the clock in X or the clock in S measures the proper time.

(iv) Explain why, according to S, the setting of the clock in X and the setting of the clock in Y does not occur simultaneously.

The spaceship S in (b) is moving with speed 0.750c as measured by X and Y and has a total energy of 2.72×10²⁰J as measured by X and Y.

(i) Determine the rest mass of spaceship S.

(ii) Using the axes, sketch a graph to show how the mass m of spaceship S changes with its speed v. Your graph should identify the rest mass m=m₀ and the speed v=c.

Muons are produced in the upper atmosphere of Earth and travel towards the surface of Earth where they are detected. Explain how, with reference to the situation described in (b), the production and detection of muons provide evidence for time dilation.

a set of coordinates that can be used to locate events/position of objects;

(i) \(\frac{{1.80 \times {{10}^{11}}}}{{0.750 \times 3 \times {{10}^8}}}\);
=800(s);
Award [2] for a bald correct answer.

(ii) \(\gamma  = \left( {\frac{1}{{\sqrt {1 - {{0.750}^2}} }} = } \right)1.51\);
\({\rm{time}} = \left( {\frac{{800}}{{1.51}} = } \right)530\left( {\rm{s}} \right)\);
Watch for ECF from (b)(i) or first marking point in (b)(ii).
Award [2] for a bald correct answer.

(iii) only S’s clock measures proper time;
because S’s clock is at both events / events occur at same place in S’s frame;

(iv) according to S, Y moves towards/X moves away from the radio signal;
the signal travels at the same speed/at the speed of light in each direction;
therefore according to S’s clock the signal reaches Y before it reaches
X/X after reaching Y;

or

S’s frame is different/moving relative to the X and Y frame;
the two events/arrival of signals are separated in space;
so if simultaneous for XY, cannot be simultaneous for S;

(i) \({\rm{rest mass}} = \frac{{{\rm{total energy}}}}{{\gamma {c^2}}}\); (allow ECF for γ from 12(b)(ii))
\( = \left( {\frac{{2.72 \times {{10}^{20}}}}{{1.51 \times 9 \times {{10}^{16}}}} = } \right)2.0 \times {10^3}\left( {{\rm{kg}}} \right)\);
Award [1 max] if gamma is not used and answer is 3000 (kg).
Award [2] for a bald correct answer.
(ii)
general shape showing asymptote to v=c;
non-zero value for m=m₀;

the muons are equivalent to S and the Earth to X,Y;
without time dilation most muons would decay before reaching the Earth’s surface;
from Earth frame, with time dilation, the (proper) half-life of muons becomes dilated/larger;
from muon frame, with time dilation, the (proper) journey time is less than that on Earth; } (do not award marking point for arguments just based on length contraction)
fewer muons decay/more muons survive than expected without time dilation; } (award marking point even if no explanation is given)