Answer to Question #98250 in Mechanics | Relativity for AbdulRehman

Question

Suppose that you are in an airplane and that you are watching another
airplane fly in the opposite direction. Your relative velocity is roughly
600m/s. Calculate the size of the time dilation effect you would observe if
you measured the ticking of a clock in the other airplane. If that clock ticksonce each second, how much time passes in your reference frame between
two of it’s ticks?
Hint: Your calculator may not be able to deal effectively with the tiny
numbers involved. As a result, if you just try to type formula (3.3) into
your calculator you may get the value 1. What I want to know is how
much the actual value is different from 1. In cases like this, it is helpful to
use Taylor series expansions. The expansions you need to complete this
problem were given in problem (2-2).

Answer 1

We can use the values expression for time dilation:

$$t=\frac{t_0}{\sqrt{1-v^2/c^2}}=\\ \space\\=\frac{1}{\sqrt{1-(600/300000000)^2}}=\\ =1.000000000002\text{ s}.$$

Yes, there are 11 zeros between 1 and 2. You can use any math processing websites or programs to get the answer less interesting but easier way than Tailor series.