91Ó°ÊÓ

How fast must a particle move before its kinetic energy equals its rest energy? [0.866 c]

A particle with four-momentum \(P\) is observed by an observer who moves with four-velocity U \(_{0}\). Prove that the energy of the particle relative to that observer is \(\mathbf{U}_{0} \cdot \mathbf{P}\).

Two particles with rest masses \(m_{1}\) and \(m_{2}\) move collinearly in some inertial frame, with uniform velocities \(u_{2}\) and \(u_{2}\), respectivcly. They collide and form a single particle with rest mass \(m\) moving at velocity \(u\). Prove that $$ m^{2}=m_{1}^{2}+m_{2}^{2}+2 m_{1} m_{2} \gamma\left(u_{1}\right) \gamma\left(u_{2}\right)\left(1-u_{1} u_{2} / c^{2}\right) $$ and also find \(u\). [Hint: for the first part, use a four-vector argument, or a result of Section \(30 .]\)

The position vector of the centre of mass of a system of particles in any inertial frame is defined by \(\mathbf{r}_{\mathrm{CM}}=\Sigma m r / \Sigma m\). If the particles suffer only collision forces, prove that \(\dot{\boldsymbol{r}}_{\mathrm{CM}}=u_{\mathrm{CM}}(\cdot \equiv \mathrm{d} / \mathrm{d} t)\); i.e. the centre of mass moves with the velocity of the CM frame. [Hint: \(\Sigma m\), \Sigmami are constant; \Sigmamir is zero between collisions, and at any collision we can factor out the r of the participating particles: \(r \Sigma \dot{m}=0 .]\)

. Show that a photon cannot spontaneously disintegrate into an electron- positron pair. [Hint: four-momentum conservation.] But in the presence of a stationary nucleus (acting as a kind of catalyst) it can. If the rest mass of the nucleus is \(N\), and that of the electron (and positron) is \(m\), what is the threshold frequency of the photon? Verify that for large \(N\) the efficiency is \(\sim 100\) per cent (cf. the preceding problem), so that the nucleus then comes close to being a pure catalyst.

A fast electron of rest mass \(m\) decelerates in a collision with a heavy nucleus and emits a (bremsstrahlung) photon. Prove that the energy of the photon can range all the way up to \((\gamma-1) m c^{2}\), the kinetic energy of the electron. [Hint: use a four-vector argument.]

A particle of rest mass \(m\) decays from rest into a particle of rest mass \(m\) ' and a photon. Find the separate energies of these end products. [Answer: \(c^{2}\left(m^{2} \pm m^{\prime 2}\right) / 2 m\). Hint: use a four-vector argument.]

15\. A rocket propels itself rectilinearly by emitting radiation in the direction opposite to its motion. If \(V\) is its final velocity relative to its initial rest frame, prove \(a b\) initio that the ratio of the initial to the final rest mass of the rocket is given by $$ \frac{M_{\mathrm{i}}}{M_{\mathrm{f}}}=\left(\frac{c+V}{c-V}\right)^{1 / 2} $$ and compare this with the result of Exercise 5 above. [Hint: equate energies and momenta at the beginning and at the end of the acceleration, writing \(\Sigma h v\) and \(\Sigma h v / c\) for the total energy and momentum, respectively, of the emitted photons.]