Section 10.7: Special Relativity (Intro)
Special relativity, proposed by Einstein in 1905, deals with the physics of objects moving at speeds close to the speed of light. It modifies classical mechanics to account for the constancy of the speed of light and time dilation effects.
- Postulates of Special Relativity:
- The laws of physics are the same in all inertial frames.
- The speed of light in vacuum is constant for all observers, regardless of the motion of the source or observer.
- Time Dilation: Moving clocks run slower: \[ t = \frac{t_0}{\sqrt{1 - v^2/c^2}} \] where \(t_0\) = proper time, \(v\) = relative velocity, \(c\) = speed of light.
- Length Contraction: Moving objects appear shorter along the direction of motion: \[ L = L_0 \sqrt{1 - v^2/c^2} \] where \(L_0\) = proper length.
- Relativistic Momentum: \[ p = \frac{mv}{\sqrt{1 - v^2/c^2}} \]
- Relativistic Energy: \[ E = \gamma m c^2 \] where \( \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} \)
Example: Time Dilation
A spaceship travels at 0.8c relative to Earth. If 5 hours pass on the spaceship, how much time passes on Earth?
\( t = \frac{t_0}{\sqrt{1 - v^2/c^2}} = \frac{5}{\sqrt{1 - 0.8^2}} \approx 8.33 \, \text{hours} \)
Practice Problems
- A rod of proper length 10 m moves at 0.6c relative to an observer. Find its contracted length.
- An astronaut travels at 0.9c for 3 years (ship time). How much time passes on Earth?
- Calculate the relativistic momentum of a 2 kg particle moving at 0.7c.
- Compute the total energy of a 1 kg object moving at 0.5c.
- Explain why no object with mass can reach the speed of light.