8. If a space ship travels at 80% the speed of light how time dilation does aff a board compared to an earth bound clock?

To calculate the time dilation factor, we can use the formula for time dilation in special relativity:<br /><br />\[ \text{Time dilation factor} = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]<br /><br />where \( v \) is the velocity of the spaceship and \( c \) is the speed of light.<br /><br />Given that the spaceship travels at \( 80\% \) the speed of light, we can substitute the values into the formula:<br /><br />\[ \text{Time dilation factor} = \frac{1}{\sqrt{1 - \left(\frac{0.8c}{c}\right)^2}} \]<br /><br />Simplifying the expression inside the square root:<br /><br />\[ \text{Time dilation factor} = \frac{1}{\sqrt{1 - 0.64}} \]<br /><br />\[ \text{Time dilation factor} = \frac{1}{\sqrt{0.36}} \]<br /><br />\[ \text{Time dilation factor} = \frac{1}{0.6} \]<br /><br />\[ \text{Time dilation factor} = \frac{5}{3} \]<br /><br />Therefore, the time dilation factor for a spaceship traveling at \( 80\% \) the speed of light is \( \frac{5}{3} \). This means that time on the spaceship will appear to move \( \frac{5}{3} \) times slower compared to an Earth-bound clock.