Chapter 15, Problem 13A

To determine

The length of a meterstick that will appear.

Answer to Problem 13A

The meterstick will appear one-tenth of its original length.

Explanation of Solution

Given:

The meterstick is thrown like spear at $99.5\%$ the speed of light

Formula used:

Length contraction can be calculated by using the formula,

  $l=l_o\sqrt{1-\frac{v^2}{c^2}}$

  $\left(1\right)$

Where, $l$ is the length of an object moving with relativistic speed,

  $l_o$ is the length of an object at rest

  $v$ is the velocity of object

  $c$ is the velocity of light

Calculation:

Length of a meterstick at rest is $l_0$

Velocity of a meterstick is $v=\frac{99.5}{100}c=0.995c$

Substituting the numerical values in equation $\left(1\right)$ ,

  $\begin{array}{c}l=l_o\sqrt{1-\frac{{\left(0.995c\right)}^2}{c^2}}\\ =l_o\sqrt{1-0.990025}\\ =l_o\sqrt{0.009975}\\ =l_o\times 0.099\\ \approx 0.1l_o\\ \approx \frac{1}{10}{l}_o\end{array}$

Conclusion:

The length of a meterstick will appear one-tenth of the original length.