What is the momentum of a proton moving at $0.987\mathrm{c}$?

(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?

An event has spacetime coordinates (x,t) = (1200 m, 2.0 μs) in reference frame S. What are the event's spacetime coordinates (a) in reference frame S' that moves in the positive x-direction at 0.80c and (b) in reference frame S'' that moves in the negative x-direction at 0.80c?

A laboratory experiment shoots an electron to the left at 0.90c. What is the electron's speed, as a fraction of c, relative to a proton moving to the right at 0.90c?

Let's examine whether or not the law of conservation of momentum is true in all reference frames if we use the Newtonian definition of momentum: p_x = mu_x. Consider an object A of mass 3m at rest in reference frame S. Object A explodes into two pieces: object B, of mass m, that is shot to the left at a speed of c/2 and object C, of mass 2m, that, to conserve momentum, is shot to the right at a speed of c/4. Suppose this explosion is observed in reference frame S' that is moving to the right at half the speed of light. Use the Lorentz velocity transformation to find the velocities and the Newtonian momenta of B and C in S'.