Homework 1 - Relatività ristretta
Completion requirements
Risolvere i seguenti esercizi/problemi:
1. Observer S assigns the following space-time coordinates to an event: x=100km, y=10km, z=55km, t=200microseconds. What are the coordinates of this event in a frame S', which moves in the direction of increasing x with speed 0.95c? Check your answers by using the inverse Lorentz transformation equations to obtain the original data in S from the results in S'.
2. A moving clock moves along the x axis at a speed of 0.60c and reads zero as it passes the origin. What time does it read as it passes the 180m mark on this axis?
3. A moving rod lies parallel to the x axis of reference frame S, moving along this axis at a speed of 0.60c. Its rest length is 1.0 m. What will be its measured length in frame S ?
4. Frame S' moves relative to frame S at 0.60c in the direction of x. In frame S' a particle is measured to have a velocity of 0.40c in the direction of increasing x'. (a) What is the velocity of the particle with respect to frame S? (b) What would be the velocity of the particle with respect to S if it moved at 0.40c in the direction of decreasing x' in the S' frame? In each case, compare your answers with the predictions of the classical velocity transformation equation.
5. At the Stanford Linear Accelerator Center (SLAC), electrons are accelerated to eneregies of 50 GeV (1 GeV = 10^9 eV). Taking the electron mass to be 0.5 MeV/c^2, calculate the corresponding momentum in GeV/c, the Lorentz factor gamma and the electrons' actual speed. Compare the resulting speed with the value you would obtain classically, if this energy were the classical kinetic energy.
6. GPS satellites move in orbits 20200 km above the surface of the earth. Assuming circular orbits, find the speed of a GPS satellite. Neglecting general relativity effects and considering the reference frames of the earth and satellite to be connected by Lorentz transformations, find the time difference between a clock in a satellite and one on the ground after a complete orbit, assuming they were initially synchronized (neglect all general relativity effects and consider only time dilation due to relative motion). Assuming we forgot to take into account this time difference, estimate the order of magnitude in the resulting error in the calculation of our position (see a brief description of the GPS system on Taylor-Zafiratos-Dubson, section 2.11, p.78).
1. Observer S assigns the following space-time coordinates to an event: x=100km, y=10km, z=55km, t=200microseconds. What are the coordinates of this event in a frame S', which moves in the direction of increasing x with speed 0.95c? Check your answers by using the inverse Lorentz transformation equations to obtain the original data in S from the results in S'.
2. A moving clock moves along the x axis at a speed of 0.60c and reads zero as it passes the origin. What time does it read as it passes the 180m mark on this axis?
3. A moving rod lies parallel to the x axis of reference frame S, moving along this axis at a speed of 0.60c. Its rest length is 1.0 m. What will be its measured length in frame S ?
4. Frame S' moves relative to frame S at 0.60c in the direction of x. In frame S' a particle is measured to have a velocity of 0.40c in the direction of increasing x'. (a) What is the velocity of the particle with respect to frame S? (b) What would be the velocity of the particle with respect to S if it moved at 0.40c in the direction of decreasing x' in the S' frame? In each case, compare your answers with the predictions of the classical velocity transformation equation.
5. At the Stanford Linear Accelerator Center (SLAC), electrons are accelerated to eneregies of 50 GeV (1 GeV = 10^9 eV). Taking the electron mass to be 0.5 MeV/c^2, calculate the corresponding momentum in GeV/c, the Lorentz factor gamma and the electrons' actual speed. Compare the resulting speed with the value you would obtain classically, if this energy were the classical kinetic energy.
6. GPS satellites move in orbits 20200 km above the surface of the earth. Assuming circular orbits, find the speed of a GPS satellite. Neglecting general relativity effects and considering the reference frames of the earth and satellite to be connected by Lorentz transformations, find the time difference between a clock in a satellite and one on the ground after a complete orbit, assuming they were initially synchronized (neglect all general relativity effects and consider only time dilation due to relative motion). Assuming we forgot to take into account this time difference, estimate the order of magnitude in the resulting error in the calculation of our position (see a brief description of the GPS system on Taylor-Zafiratos-Dubson, section 2.11, p.78).
Last modified: Monday, 28 October 2013, 6:36 PM