<p>University of Bahrain PHYCS 222 Modern Physics Department of Physics Dr. M. El-Hilo First Semester 2011/2012 Ch. 2: Special theory of relativity Task 1: Graph the variation of (v)=1/(1-(v/c)2)1/2 vs. v/c, what do you understand from the variation of  with speed?. </p><p>Task 2: Graph the Galilean (non-relativistic) and the Lorentz (relativistic) velocities, ux, relative to a fixed system for an object moving with ux=0.5c. What can you conclude from these variations? </p><p>Task 3: Graph the decay of Muon with travelled distance; consider the followings: </p><p>N0=1000, Muon’s speed=0.98c, half-life as seen by the Muon is 1.52s, half-life time as seen by us is 6.8s. Sketch the distance travelled by the Muon as seen by the Muon and as seen by us. What are the proper time and length in this problem. Why?. </p><p>1 Task 4 Relativistic Dynamics: Velocities and Mass </p><p>Consider the following collision that is happing in the fixed frame: A particle of mass m is moving along the positive x-axis at constant speed u1x=0.8c. Another particle of the same mass is moving along the positive y-axis at a constant speed u1y=0.6c. After they collide they become one mass of 2m and move at a speed u1f. Using the non-relativistic momentum conservation calculations you will find that u1f=0.5c since u1xf=0.4c and u1yf=0.3c. So in the K system, the collision appears as follow</p><p>Collision as seen in the Lab, Non relativistic calculations Before Collision After collision u1xi u1yi u1 u2xi u2yi u2 uxf uyf uf 0.8c 0 0.8c 0 0.6c 0.6c 0.4c 0.3c 0.5c</p><p>px=2m*0.4c-m*0.8c=0!!</p><p>py=2m*0.3c-m*0.6c=0!! Now, a space ship is moving along the positive x-axis at v=0.866c and the astronaut witness the same collision. Use relativistic velocities and describe how the collision look like as seen by the astronaut, i.e what are the velocities before and after the collision as seen by him (Sketch the collision as seen by the astronaut). After you find u1xi, u1yi, u1i, u2xi, u2yi, u2i, uxf, uyi, and uf then ask the astronaut to check the conservation of momentum, i.e. let him calculate px and </p><p>py, what did he find, which postulate does your answer violates? List all your results for the velocities and speed before and after collision as seen by the astronaut in a table similar to the above table. </p><p>Collision as seen by someone moving to the right at v=0.866c Before Collision After collision u1xi u1yi u1 u2xi u2yi u2 uxf uyf uf</p><p> p1xi p1yi p2xi p2yi pxf pyf pf</p><p>pxi= pyi= pxf= pyf= </p><p>2 Repeat all you previous calculations, in both the fixed and moving systems using both relativistic velocities and mass. Do you get in both frames that the momentum is conserved? </p><p>Collision as seen in the Lab, relativistic calculations Before Collision After collision u1xi u1yi u1 u2xi u2yi u2 uxf uyf uf</p><p> p1xi p1yi p2xi p2yi pxf pyf</p><p>pxi= pyi= pxf= pyf= </p><p>Collision as seen by someone moving to the right at v=0.866c Before Collision After collision u1xi u1yi u1 u2xi u2yi u2 uxf uyf uf</p><p> p1xi p1yi p2xi p2yi pxf pyf</p><p>pxi= pyi= pxf= pyf= </p><p>Task 5: Relativistic Energy</p><p>Plot the variation of the kinetic energy (K/E0) versus u/c for both the non- relativistic and relativistic cases. What do you conclude from this? Determine at what value of u/c the percentage error in using the non-relativistic equation for the kinetic energy is equal 1%. An electron is accelerated using a potential difference of 1000000V, what will be the error in you calculations for the final speed if you use the non-relativistic equation for the kinetic energy?</p><p>You did the e/m experiment and a potential difference of VA= 5000 V was used to accelerate the electrons. Are you okay in this experiment with the non-relativistic equation for the kinetic energy? </p><p>3 Task 6: Task 6: Relativistic energy and momentum</p><p>An electron, photon and proton each has a momentum of 4GeV/c. For each particle find its total energy, kinetic energy and speed. List your results in the following table. What can you conclude from your calculations when Pc>E0?</p><p>In a system that is fixed relative to the three particles Particle P(GeV/c) E (GeV) KE (GeV) u/c Photon 4 Electron 4 Proton 4</p><p>The three particles are studied by someone in a moving system. In what moving system, the proton will appear to have a momentum of 1GeV/c. In that frame of reference, find the particle’s momentum, total energy, kinetic energy and speed relative to the moving system. List your results in the following table. </p><p>In a system that is moving relative to the three particles (v= ) Particle P(GeV/c) E (GeV) KE (GeV) u/c Photon Electron Proton 1</p><p>What can you conclude from all these calculations? What postulates that are satisfied in these calculations? </p><p>4 Task 7: All Relativistic </p><p>In a laboratory frame S, a proton of kinetic energy equals twice its rest energy</p><p>(i.e. Kp=2Eop) is moving along the positive x-axis collided with another proton initially at rest. If we assume that after the collision, the incoming proton is stopped and the resting proton move along the +ve x-axis. This collision is also studied by an Alien moving along the +ve x-axis at speed v= 0.866c relative to the laboratory frame. </p><p>Lab </p><p>Calculate the followings and list them in the following table: </p><p>Relative to the laboratory frame (S) Relative to the Alien Total Kinetic Momentu Velocity Total Kinetic Momentu Velocity Energy Energy m Energy Energy m (GeV) (GeV) (GeV/c) (GeV) (GeV) (GeV/c) Incident proton Resting proton E , P before collision Stopped proton Ejected proton E , P after collision</p><p>5</p>