(I) An observer on Earth sees an alien vessel approach at a speed of 0.70c. The fictional starship Enterprise comes to the rescue (Fig. 36–17), overtaking the aliens while moving directly toward Earth at a speed of 0.90c relative to Earth. What is the relative speed of one vessel as seen by the other?

In the old West, a marshal riding on a train traveling 35.0 m/s sees a duel between two men standing on the Earth 55.0 m apart parallel to the train. The marshal’s instruments indicate that in his reference frame the two men fired simultaneously.

(a) Which of the two men, the first one the train passes (A) or the second one (B) should be arrested for firing the first shot? That is, in the gunfighter’s frame of reference, who fired first?

(b) How much earlier did he fire?

(c) Who was struck first?

An observer in reference frame S notes that two events are separated in space by 180 m and in time by 0.80μs. How fast must reference frame S' be moving relative to S in order for an observer in S' to detect the two events as occurring at the same location in space?

Using Example 36–2 as a guide, show that for objects that move slowly in comparison to c, the length contraction formula is roughly ℓ ≈ ℓ₀ (1 - 1/2 v²/c²) . Use this approximation to find the “length shortening” ∆ℓ = ℓ₀ - ℓ of the train in Example 36–6 if the train travels at 100 km/h (rather than 0.92c).

What magnetic field B is needed to keep 998-GeV protons revolving in a circle of radius 1.0km? Use the relativistic mass. The proton’s “rest mass” is 0.938 GeV/c². ( 1 GeV = 10⁹ eV.) [Hint: In relativity, mᵣₑₗ v²/r = qvB is still valid in a magnetic field, where mᵣₑₗ = γm.]

(III) If a particle moves in the xy plane of system S (Fig. 36–12) with speed u in a direction that makes an angle θ with the x axis, show that it makes an angle θ' in S' given by $\tan {\theta }^{\prime }=\left(\sin \theta \right)\sqrt{1-v^2/c^2}/(\cos \theta -v/u)$.

(III) (a) In reference frame S, a particle has momentum $\overrightarrow{p}=p_{x}i$ along the positive x axis. Show that in frame S’, which moves with speed v as in Fig. 36–12, the momentum has components

$p_x^{\prime }=\frac{px-vE/c^2}{\sqrt{1-v^2/c^2}}$

$p_y^{\prime }=py$

$p_z^{\prime }=pz$

$E^{\prime }=\frac{E-p_{x}v}{\sqrt{1-v^2/c^2}}.$

(These transformation equations hold, actually, for any direction of $\overrightarrow{p}$, as long as the motion of S' is along the x axis.) (b) Show that p_x, p_y, p_z, E/c transform according to the Lorentz transformation in the same way as x, y, z, ct.

In a lab frame, S, an object crosses a distance of 15 m in 10 s. In an initially aligned frame S', moving at 1000 km/s in the x-direction relative to S, how far a distance does the object have to travel, and in what time does it travel the distance?

In a particle accelerator, a neutron is traveling at a speed of 0.7 c, as measured by you in a laboratory. This neutron decays (becoming a proton), ejecting an electron. If you measure the electron's speed to be 0.5 c, traveling in the same direction as the neutron, what was the relative speed between the electron and neutron when the neutron decayed? Was the electron ejected forward or backwards relative to the neutron's motion, as 'seen' by the neutron?