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## Relativistic Mechanics ### Time Dilation Moving clocks run slow. If $\Delta t_0$ is the time between two events that occur at the same location, then the time $\Delta t$ between those events as measured in a moving frame is: $\qquad \Delta t = \gamma \Delta t_0 \qquad (1)$ where $\qquad \gam...

## Relativistic Mechanics ### Time Dilation Moving clocks run slow. If $\Delta t_0$ is the time between two events that occur at the same location, then the time $\Delta t$ between those events as measured in a moving frame is: $\qquad \Delta t = \gamma \Delta t_0 \qquad (1)$ where $\qquad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$ * $\Delta t_0$ is called the proper time. * The effect is called time dilation. ### Length Contraction Moving objects are shorter than when they are at rest. If $L_0$ is the length of an object at rest, then the length $L$ when the object moves along the direction of the length is $\qquad L = \frac{L_0}{\gamma} \qquad (2)$ * $L_0$ is called the proper length. * The effect is called length contraction. * Length contraction happens only along the direction of motion. Lengths perpendicular to the direction of motion are not contracted. ### Velocity Transformation A spaceship moving relative to you at velocity $v$ fires a missile with velocity $u'$ relative to the spaceship. Then the velocity $u$ of the missile relative to you is $\qquad u = \frac{u' + v}{1 + \frac{u'v}{c^2}} \qquad (3)$ If both $u'$ and $v$ are much smaller than $c$, then the term $\frac{u'v}{c^2}$ is close to zero, and the equation becomes the familiar (Galilean) velocity transformation $u = u' + v$. ### Relativistic Momentum The relativistic momentum of a particle with mass $m$ moving at velocity $v$ is: $\qquad p = \gamma mv \qquad (4)$ ### Relativistic Energy The relativistic energy of a particle with mass m moving at velocity $v$ is: $\qquad E = \gamma mc^2 \qquad (5)$ The relativistic kinetic energy of a particle with mass $m$ moving at velocity $v$ is: $\qquad K = E - mc^2 = (\gamma - 1) mc^2 \qquad (6)$ ### Massless Particles Particles with zero mass (such as photons) must travel at the speed of light. Their energy and momentum are related by $\qquad E = pc \qquad (7)$ ### Relation Between Energy and Momentum Energy and momentum are related by $\qquad E^2 = (pc)^2 + (mc^2)^2 \qquad (8)$ ### What is the same in all frames? The quantity $E^2 - (pc)^2$ is the same in all frames.