Two particles are created in a high-energy accelerator and move off in opposite directions. The speed of one particle, as measured in the laboratory, is 0.650c, and the speed of each particle relative to the other is 0.950c. What is the speed of the second particle, as measured in the laboratory?

(I) Repeat Problem 19 using the Lorentz transformation and a relative speed v = 1.60 x 10⁸ m/s, but choose the time t to be (a) 3.5μs and (b) 10.0 μs .

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?

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.]

(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?

(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.