Homework

For the purposes of brevity and clarity, let us define the inertial frame $K$ to be at rest in the lab with coordinates $x_\mu$ and define another inertial frame $K'$ to coincide with $K$ at $t'=t=0$ and to be moving relative to $K$ with a velocity $v=\beta c$ in the x direction.

1. Assume an object of length $\ell$ is at rest in the $K$ frame. Show that the expression for the Lorentz contraction can be obtained if an observer in the $K'$ frame measures the time it takes for an object to pass and multiplies by the velocity $v$.
2. Two events take place at the same time but at positions differing by $\Delta x$ in the $K$ frame. Calculate the difference in position $\Delta x'$ and the difference in time $\Delta t'$ in the $K'$ frame.
3. Clocks located at the origins of the $K$ and $K'$ frames are synchronized at $t=0$. At a time $t$, an observer at the origin of the $K$ system uses a telescope to read the the clock moving with the $K'$ system. What time does he read.
4. A relativistic rocket emits exhaust gasses with a constant velocity $v_e$ in the rest frame of the rocket and at a rate $-{dm\over dt}$. Show that the equation of motion for the rocket is $m(t){dv\over dt}+v_e{dm\over dt}(1-\beta^2)^{3/2}=0$, where $m(t)$ is the mass of the rocket.
5. A muon is an unstable particle with a mean lifetime of 2 microseconds in its rest frame and a mass $mc^2=106$ MeV. What is its mean lifetime in the frame of the earth if its energy is $E=106$ GeV.
6. A nearby star is found to be moving away from the earth with a velocity of $5\times 10^4$ m/s by measuring the Balmer $\alpha$ (H-alpha) wavelength normally $\lambda=656.28$ nm. How much and in what directions is the wavelength shifted?
7. An astronaut travels to a nearby star system, a distance of 11 light years away, and returns. Both legs of the trip are made at a velocity of $0.6c$. How much have the astronaut and his twin on earth aged?
8. The total solar radiation at the position of the earth ( $1.5\times 10^{11}$ m from the sun) is measured by satellite to be 1,366 W/m$^2$. How much mass per second is converted to energy in the sun?
9. What is the kinetic energy of a particle of mass $m$ and momentum $p$?
10. LEP collided electrons with positrons with a center of mass energy of 208 GeV. The two particles have the same energy (104 GeV) in the lab. What is $\beta$ for these particles?
11. A neutral pion ($mc^2=135$ MeV) has a energy of 40 GeV. It decays into two photons. If the two photons have the same energy in the lab frame, what is the angle between them?
12. A neutron ( $mc^2=939.6$ MeV) decays into a proton ( $mc^2=938.3$ MeV) and an electron ( $mc^2=0.511$ MeV) and an antineutrino ( $mc^2\approx 0$ MeV). The decay is symmetric so that the three final state particles lie in a plane separated by angles of 120 degrees. What is the momentum of each particle? What is the kinetic energy of each particle? What trend do you notice?
13. Use the Lorentz transformation to derive the velocity addition formula.
14. A spacecraft passes Neptune with a speed of $0.9c$ relative to the planet. A second spacecraft, more urgently exploring the outer solar system, is observed to pass the first one (in exactly the same direction) at a speed of $0.4c$. What is the speed of the second spacecraft relative to Neptune?
15. Show that the relativisticly corrected form of Newton's Second Law is.
    $$F=m{dv\over dt}\left(1-{v^2\over c^2}\right)^{-{3\over 2}}$$

16. Calculate the kinetic energy of a particle scattered through an angle $\theta$ by a target particle of equal mass. Assume the target particle is at rest and that the incident particle has kinetic energy $T_0$.
17. Calculate the dot product of the velocity 4-vectors for two particles, $a$ and $b$. Show how it depends on the relative velocity of the two particles.
18. In modern High Energy physics, most particle accelerators collide beams of equal energy particles rather than collide a beam with a target that is at rest. Assume we want to collide protons with protons. Calculate a formula for the ``Center of Mass'' energy of the collisions for the case of a fixed target. The Large Hadron Collider will have a center of mass energy of 14,000 GeV. What energy beam would be needed to achieve this center of mass energy by colliding with a fixed target of protons (Liquid Hydrogen)?
19. Assume a neutrino beam is made by generating a beam of $\pi^+$ with kinetic energy $T=500$ GeV. The $\pi^+$ decay to a positive muon and a neutrino of the muon type, $\pi^+\rightarrow \mu^+\ \nu_\mu$. What is the energy range of neutrinos in the beam if we define the ``beam'' to be in the direction of the original pion beam within and small angle of $\Delta\theta=10^{-3}$ radians?
20. Show that Maxwell's equations written in 3D and in Rationalized Heavyside-Lorentz Units, can all be derived from the 4D Maxwell's equation in conjunction with the definition of $F_{\mu\nu}={\partial A_\nu\over \partial x_\mu}-{\partial A_\mu\over \partial x_\nu}$.
21. Assume the relativistic motion of a particle in 4D is simply given by $\delta s^2=\delta \int\limits_{-c^2\tau^2_i}^{-c^2\tau^2_f}ds^2=0$. Show that this leads to the standard non-relativistic Lagrange equation for a particle with no forces acting on it.