5.4 Length contraction  (Page 3/6)

Calculating length contraction

Strategy

Solution for (a)

1. Identify the knowns: $L_0=4.300\text{ly;}\gamma =30.00.$
2. Identify the unknown: L .
3. Express the answer as an equation: $L=\frac{L_0}{\gamma }.$
4. Do the calculation:
   
   $\begin{array}{cc}\hfill L& =\frac{L_0}{\gamma }\hfill \\ & =\frac{4.300\text{ly}}{30.00}\hfill \\ & =0.1433\text{ly.}\hfill \end{array}$

Solution for (b)

1. Identify the known: $\gamma =30.00.$
2. Identify the unknown: v in terms of c .
3. Express the answer as an equation. Start with:
   
   $\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}.$
   
   Then solve for the unknown v/c by first squaring both sides and then rearranging:
   
   $\begin{array}{ccc}\hfill {\gamma }^2& =\hfill & \frac{1}{1-\frac{v^2}{c^2}}\hfill \\ \hfill \frac{v^2}{c^2}& =\hfill & 1-\frac{1}{{\gamma }^2}\hfill \\ \hfill \frac{v}{c}& =\hfill & \sqrt{1-\frac{1}{{\gamma }^2}}.\hfill \end{array}$
4. Do the calculation:
   
   $\begin{array}{cc}\hfill \frac{v}{c}& =\sqrt{1-\frac{1}{{\gamma }^2}}\hfill \\ & =\sqrt{1-\frac{1}{{\left(30.00\right)}^2}}\hfill \\ & =0.99944\hfill \end{array}$
   
   or
   
   $v=0.9994c.$

Significance

People traveling at extremely high velocities could cover very large distances (thousands or even millions of light years) and age only a few years on the way. However, like emigrants in past centuries who left their home, these people would leave the Earth they know forever. Even if they returned, thousands to millions of years would have passed on Earth, obliterating most of what now exists. There is also a more serious practical obstacle to traveling at such velocities; immensely greater energies would be needed to achieve such high velocities than classical physics predicts can be attained. This will be discussed later in the chapter.

Why don’t we notice length contraction in everyday life? The distance to the grocery store does not seem to depend on whether we are moving or not. Examining the equation $L=L_0\sqrt{1-\frac{v^2}{c^2}},$ we see that at low velocities $(v\text{<}\text{<}c),$ the lengths are nearly equal, which is the classical expectation. But length contraction is real, if not commonly experienced. For example, a charged particle such as an electron traveling at relativistic velocity has electric field lines that are compressed along the direction of motion as seen by a stationary observer ( [link] ). As the electron passes a detector, such as a coil of wire, its field interacts much more briefly, an effect observed at particle accelerators such as the 3-km-long Stanford Linear Accelerator (SLAC). In fact, to an electron traveling down the beam pipe at SLAC, the accelerator and Earth are all moving by and are length contracted. The relativistic effect is so great that the accelerator is only 0.5 m long to the electron. It is actually easier to get the electron beam down the pipe, because the beam does not have to be as precisely aimed to get down a short pipe as it would to get down a pipe 3 km long. This, again, is an experimental verification of the special theory of relativity.