Time Dilation Worked Examples

__The Time Dilation Equation.__

Of all the major advances in physics from about 1900 onwards special relativity is the only one that can be understood in its entirety without recourse to mathematics beyond that of high school level. However, like all physics special relativity has at its base a precise set of mathematical formulas from which predictions can be made and tested against experimental results. It will come as no surprise then that time dilation has a precise mathematical formula. This is it:

__Example 1: Solving the Equation as a Factor of 1.__

The effects of time dilation don't become really noticeable until very high speeds are reached so for this worked example I will use a speed of 90% of that of light, that is 270,000 km per second (the speed of light is very close to 300,000 km per second). The first thing we must do is to write down the equation:

We now need to "plug in the numbers". Because *V*^{2}/*c*^{2}
is a ratio we can either use the exact values or just the percentages of
each value. It is easier to do the latter. Because we are only
interested in the dilated time factor we can set the stationary time to
be 1. Note that we can drop the percentage symbols and that *c*
is equal to 100% of the speed of light. Plugging in the numbers we get:

We can now begin to solve the equation. The first thing we can do is remove the leading number 1 (anything multiplied by 1 is itself), then square the last two terms:

We can now reduce the equation by carrying out the division:

We then carry out the subtraction:

And finally we take the square root (and round the answer to a workable value):

This result means that at 90% of the speed of light local time has slowed down to 43.6% of that relative to an external observer. Put another way, if a rocket is sent out into space on a 10 (Earth) year mission at 90% of the speed of light the rocket and its occupants will have only aged by about four and a half years when they return, while everyone and everything on Earth will have aged 10 years.

__Example 2: Solving the Equation as a Measure of Time.__

The first example solved the equation as a factor of one. This example puts a real time scale into the equation. In this example we will look at how time changes over 10 years travelling at a speed of 50% of that of light.

Instead of the step-by-step approach of the last example I will just carry out the equation:

So for a rocket travelling at 50% of the speed of light 8.66 years will pass in the same time as 10 years pass for a stationary observer. Note that while the unit of time used was years it could have been any other unit of time, such as seconds or millennia, as long as the result is in the same units as the units used in the equation.