Problem 50
For any two objects \(a\) and \(b\), show that the scalar product of their four- velocities is \(u_{a} \cdot u_{b}=\) \(-c^{2} \gamma\left(v_{\mathrm{rel}}\right),\) where \(\gamma(v)\) denotes the usual \(\gamma\) factor, \(\gamma(v)=1 / \sqrt{1-v^{2} / c^{2}},\) and \(v_{\mathrm{rel}}\) denotes the speed of \(a\) in the rest frame of \(b\) or vice versa.
Problem 55
Since the four-velocity \(u=\gamma(\mathbf{v}, c)\) is a four-vector its transformation properties are simple. Write down the standard Lorentz boost for all four components of \(u\). Use these to deduce the relativistic velocity- addition formula for v.
Problem 57
When a radioactive nucleus of astatine 215 decays at rest, the whole atom is torn into two in the reaction $$^{215} \mathrm{At} \rightarrow^{211} \mathrm{Bi}+^{4} \mathrm{He}$$ The masses of the three atoms are (in order) \(214.9986,210.9873,\) and \(4.0026,\) all in atomic mass units. (1 atomic mass unit \(=1.66 \times 10^{-27} \mathrm{kg}=931.5 \mathrm{MeV} / c^{2}\).) What is the total kinetic energy of the two out coming atoms, in joules and in MeV?
Problem 58
(a) What is a particle's speed if its kinetic energy \(T\) is equal to its rest energy? ( \(\mathbf{b}\) ) What if its energy \(E\) is equal to \(n\) times its rest energy?
Problem 59
If one defines a variable mass \(m_{\text {var }}=\gamma m\), then the relativistic momentum \(\mathbf{p}=\gamma m \mathbf{v}\) becomes \(m_{\text {var }} \mathbf{v}\) which looks more like the classical definition. Show, however, that the relativistic kinetic energy is not equal to \(\frac{1}{2} m_{\mathrm{var}} v^{2}\)
Problem 60
A particle of mass \(m_{a}\) decays at rest into two identical particles each of mass \(m_{b} .\) Use conservation of momentum and energy to find the speed of the outgoing particles.
Problem 61
A particle of mass 3 MeV/c \(^{2}\) has momentum 4 MeV/c. What are its energy (in MeV) and speed (in units of \(c\) )?
Problem 62
A particle of mass 12 MeV/c \(^{2}\) has a kinetic energy of 1 \(\mathrm{MeV}\). What are its momentum (in MeV/c) and its speed (in units of \(c\) )?
Problem 63
(a) What is a mass of \(1 \mathrm{MeV} / c^{2}\) in kilograms? ( \(\mathbf{b}\) ) What is a momentum of \(1 \mathrm{MeV} / c\) in \(\mathrm{kg} \cdot \mathrm{m} / \mathrm{s} ?\)
Problem 64
As measured in the inertial frame \(\mathcal{S},\) a proton has four-momentum \(p\). Also as measured in \(\mathcal{S}\), an observer at rest in a frame \(\mathcal{S}^{\prime}\) has four-velocity \(u\). Show that the proton's energy, as measured by this observer, is - \(u \cdot p\).
