Chapter 15, Problem 14A

To determine

The length of a meterstick would appear if it were thrown like spear at $99.5\%$ the speed of light, but with its length perpendicular to its direction of motion.

Answer to Problem 14A

  $l=1\text{ m}$

Explanation of Solution

Introduction:

According to Einstein’s special theory of relativity - for an observer, an object moving with a relativistic speed is looks like, it is of small length compared to its actual length. This happens only in one dimension. The quantity of this length contraction is entirely governed by the speed of a body relative to an observer. Length contraction can be expressed as,

  $l=l_o\sqrt{1-\frac{v^2}{c^2}}$

Where, $l$ is the length of a body moving with relativistic speed,

  $l_o$ is the length of a body at rest

  $v$ is the velocity of a body

  $c$ is the velocity of light

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

According to Einstein’s special theory of relativity, the length contraction happens merely in one direction and is in the direction of motion of the object. Hence the meterstick would not appear to be contracted in the direction perpendicular to its direction of motion. Length of a meterstick is $1\text{ m}$ . Hence if the meterstick were thrown like a spear with a speed $0.99c$ , the length of a meterstick would appear in the direction perpendicular to its direction of motion is $1\text{ m}$ .

Conclusion:

There is no change is observed in the length of a meterstick perpendicular to its direction of motion.