IMG_1497.jpeg

# Chapter 14: The Special Theory of Relativity ## 14.1 The Postulates of Special Relativity ### Galilean Relativity * Classical relativity principle: The laws of physics are the same in all inertial reference frames. * Inertial reference frame: A frame in which an isolated object has constant...

# Chapter 14: The Special Theory of Relativity ## 14.1 The Postulates of Special Relativity ### Galilean Relativity * Classical relativity principle: The laws of physics are the same in all inertial reference frames. * Inertial reference frame: A frame in which an isolated object has constant velocity. ### The Michelson-Morley Experiment * Search for the "ether," the medium through which light was thought to propagate. * Experiment showed that the speed of light is the same in all directions, independent of the Earth's motion. ### Einstein's Postulates 1. The laws of physics are the same in all inertial reference frames. 2. The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source. ## 14.2 The Relativity of Time and Length ### The Relativity of Time * Thought experiment: A light clock in two different reference frames. * Time dilation: Time intervals are longer in a moving reference frame. * $\Delta t = \gamma \Delta t_0$ * $\Delta t_0$ is the proper time (shortest possible time interval between two events) * $\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$ * v is the relative speed of the two reference frames. ### The Relativity of Length * Length contraction: Lengths are shorter in a moving reference frame. * $L = L_0/\gamma$ * $L_0$ is the proper length (longest possible length of an object) ### Example * A spaceship travels at 0.8c relative to Earth. * A clock on the spaceship ticks for 1 hour. How long does this take as measured by an observer on Earth? * The spaceship is 100 m long as measured by observers on the spaceship. How long is the spaceship as measured by an observer on Earth? ### Answer * Time Dilation: * $\gamma = 1/\sqrt{1 - v^2/c^2} = 1/\sqrt{1 - 0.8^2} = 5/3$ * $\Delta t = \gamma \Delta t_0 = (5/3)(1 \text{ hour}) = 1.67 \text{ hours}$ * Length Contraction: * $L = L_0/\gamma = (100 \text{ m})/(5/3) = 60 \text{ m}$ ## 14.3 Relativistic Momentum and Energy ### Relativistic Momentum * $\vec{p} = \gamma m\vec{v}$ ### Relativistic Energy * $E = \gamma mc^2 = KE + mc^2$ * Kinetic energy: $KE = (\gamma - 1)mc^2$ * Rest energy: $E_0 = mc^2$ ### Massless Particles * $E = pc$ ### Example * What is the kinetic energy of an electron moving at 0.9c? * What is the momentum of a photon with energy 1 eV? ### Answer * Kinetic Energy: * $\gamma = 1/\sqrt{1 - v^2/c^2} = 1/\sqrt{1 - 0.9^2} = 2.29$ * $KE = (\gamma - 1)mc^2 = (2.29 - 1)(9.11 \times 10^{-31} \text{ kg})(3.00 \times 10^8 \text{ m/s})^2 = 1.17 \times 10^{-13} \text{ J}$ * Momentum: * $E = 1 \text{ eV} = 1.60 \times 10^{-19} \text{ J}$ * $p = E/c = (1.60 \times 10^{-19} \text{ J})/(3.00 \times 10^8 \text{ m/s}) = 5.33 \times 10^{-28} \text{ kg m/s}$