Define proper length.

A spaceship is travelling to the right at speed 0.75 c, through a tunnel which is open at both ends. Observer A is standing at the centre of one side of the tunnel. Observer A, for whom the tunnel is at rest, measures the length of the tunnel to be 240 m and the length of the spaceship to be 200 m. The diagram below shows this situation from the perspective of observer A.

Observer B, for whom the spaceship is stationary, is standing at the centre of the spaceship.

(i) Calculate the Lorentz factor, γ, for this situation.

(ii) Calculate the length of the tunnel according to observer B.

(iii) Calculate the length of the spaceship according to observer B.

(iv) According to observer A, the spaceship is completely inside the tunnel for a short time. State and explain whether or not, according to observer B, the spaceship is ever completely inside the tunnel.

Two sources of light are located at each end of the tunnel. The diagram below shows this situation from the perspective of observer A.

According to observer A, at the instant when observer B passes observer A, the two sources of light emit a flash. Observer A sees the two flashes simultaneously. Discuss whether or not observer B sees the two flashes simultaneously.

the length of an object as measured by an observer who is at rest relative to the object;

(i) \(\gamma  = \frac{1}{{\sqrt {1 - \frac{{{v^2}}}{{{c^2}}}} }} = \frac{1}{{\sqrt {1 - {{0.75}^2}} }} = 1.5\);

(ii) \(L = \frac{{{L_0}}}{\gamma } = \frac{{240}}{{1.5}} = 160{\rm{m}}\);

(iii) \({L_0} = \gamma L = 1.5 \times 200 = 300{\rm{m}}\); 

(iv) the spaceship is never completely inside the tunnel;
because (according to observer B) the spaceship is longer than the tunnel;
Apply ECF in all parts of question (b).

observer B will not see the two flashes simultaneously;
according to B, light 2 is moving to the left/towards observer B;
since the speed of light is the same for both sources;
the flash from light 2 reaches B before the flash from light 1;

or

according to B, the two flashes arrive at A simultaneously;
according to B, A is moving to the left/away from light 2;
since light from both sources moves with the same speed;
for the flashes to be received by A at the same time, the flash from light 2 must be emitted first; 
Accept any equivalent discussion.