Chapter 35

An electron's rest mass is \(0.511 \mathrm{MeV} / \mathrm{c}^{2}\) a) How fast must an electron be moving if its energy is to be 10 times its rest energy? b) What is the momentum of the electron at this speed?

The Relativistic Heavy Ion Collider (RHIC) can produce colliding beams of gold nuclei with beam kinetic energy of \(A \cdot 100 .\) GeV each in the center-of- mass frame, where \(A\) is the number of nucleons in gold (197). You can approximate the mass energy of a nucleon as approximately \(1.00 \mathrm{GeV}\). What is the equivalent fixed-target beam energy in this case?

How much work is required to accelerate a proton from rest up to a speed of \(0.997 c ?\)

In proton accelerators used to treat cancer patients, protons are accelerated to \(0.61 c\). Determine the energy of the proton, expressing your answer in MeV.

In some proton accelerators, proton beams are directed toward each other for head-on collisions. Suppose that in such an accelerator, protons move with a speed relative to the lab of \(0.9972 c\). a) Calculate the speed of approach of one proton with respect to another one with which it is about to collide head on. Express your answer as a multiple of \(c\), using six significant digits. b) What is the kinetic energy of each proton beam (in units of \(\mathrm{MeV}\) ) in the laboratory reference frame? c) What is the kinetic energy of one of the colliding protons (in units of \(\mathrm{MeV}\) ) in the rest frame of the other proton?

The hot filament of the electron gun in a cathode ray tube releases electrons with nearly zero kinetic energy. The electrons are next accelerated under a potential difference of \(5.00 \mathrm{kV}\), before being steered toward the phosphor on the screen of the tube. a) Calculate the kinetic energy acquired by the electron under this accelerating potential difference. b) Is the electron moving at relativistic speed? c) What is the electron's total energy and momentum? (Give both values, relativistic and nonrelativistic, for both quantities.)

Consider a one-dimensional collision at relativistic speeds between two particles with masses \(m_{1}\) and \(m_{2}\). Particle 1 is initially moving with a speed of \(0.700 c\) and collides with particle \(2,\) which is initially at rest. After the collision, particle 1 recoils with speed \(0.500 c\), while particle 2 starts moving with a speed of \(0.200 c\). What is the ratio \(m_{2} / m_{1} ?\)

In an elementary-particle experiment, a particle of mass \(m\) is fired, with momentum \(m c\), at a target particle of mass \(2 \sqrt{2} m\). The two particles form a single new particle (completely inelastic collision). Find: a) the speed of the projectile before the collision b) the mass of the new particle c) the speed of the new particle after the collision

Show that momentum and energy transform from one inertial frame to another as \(p_{x}^{\prime}=\gamma\left(p_{x}-v E / c^{2}\right) ; p_{y}^{\prime}=p_{y}\) \(p_{z}^{\prime}=p_{p} ; E^{\prime}=\gamma\left(E-v p_{x}\right) .\) Hint: Look at the derivation for the space-time Lorentz transformation.

Show that \(E^{2}-p^{2} c^{2}=E^{2}-p^{2} c^{2},\) that is, that \(E^{2}-p^{2} c^{2}\) is a Lorentz invariant. Hint: Look at derivation showing that the space-time interval is a Lorentz invariant.