Question

A radar antenna is rotating and makes one revolution every 24 s, as measured on earth. However, instruments on a spaceship moving with respect to the earth at a speed v measure that the antenna makes one revolution every 44 s. What is the ratio v/c of the speed v to the speed c of light in a vacuum

Answer 1

0.838

Step-by-step explanation:

The ratio v/c of the speed v to the speed c of light in a vacuum is shown below:

Given that

`\triangle t_0 = 24\ seconds` = time interval for one revolution

`\triangle t = 44\ seconds` = time interval measured with speed v

based on the given information, the ratio v/c of the speed v to the speed c of light in a vacuum is

`\triangle t = \frac{\triangle t_0}{\sqrt{1 - (v^2)/(c^2)}}`

`{\sqrt{1 - (v^2)/(c^2)}} = (\triangle t_0)/(\triangle t)`

Now squaring both the sides

`(v^2)/(c^2) = 1 - ((\triangle t_0)^2)/((\triangle t)^2)`

Now remove the squaring root from both the sides and putting the values

`(v)/(c) = {\sqrt{1 - ((\triangle t_0)^2)/((\triangle t)^2)`

`= {\sqrt{1 - ((24)^2)/((44)^2)`

= 0.838