Chapter 16, Problem 22A

To an Earth observer, metersticks on three spaceships are seen to have these lengths. Rank the speeds of the spaceships relative to Earth from highest to lowest.

To determine

To rank:The speed of the given spaceships relative to Earth from highest to lowest.

Answer to Problem 22A

  $C,B,A$

Explanation of Solution

Introduction:

Einstein’s special theory of relativity says that, the length of an object moving with a relativistic speed appear to be contracted along the direction of its motion. According to this, the observer at rest would perceive the length of an object to be shorter than the actual length of that object. The amount of contraction in length of an object moving with relativistic speed entirely depends on the speed of an object relative to an observer as shown in the expression,

  $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 an object

  $c$ is the velocity of light

The expression for length contraction (equation 1) clearly indicates that the object moving with large velocity appear to be smaller than that of moving with small velocity.

Consider the given diagram.

  

The meterstick in the spaceship C appear to be smallest compared to that in spaceships A and B. This shows that the spaceship C moves with largest velocity. Hence the given spaceships can be ranked according to their speed relative to Earth from highest to lowest as $C,B,A$ .

Conclusion:

The speeds of the given spaceships relative to Earth from highest to lowest is ranked as $C,B,A$ .