The best way to deal with relativity (or any nonintuitive physical system) is to deal with the math. Because the ideas of relativity at extreme speeds are foreign to us (we live our lives at low speed), mental visualization isn’t always accurate. However, it’s also true that one of the best ways to understand how something works is to play with it.
To that end, I’ve made a relativity calculator. The code is a little bit ugly, but it’s publicly available in this gist. It calculates the Lorentz transformations for mass, length, and time. Because the scales here are so vastly different (between everyday human speed and near-light speed), it was tempting to use a single log slider, but I instead opted for three different velocity sliders, because I think human intuition is really only good for linear scales.
The macro slider changes in increments of one percent of the speed of light. The medium slider has its zero point at zero, and its maximum point at one percent of the speed of light. Finally, the micro slider is adjustable in single meters per second, up to one kilometer per second. One meter per second is a pretty reasonable base speed. You’ll likely be able to approximate it pretty well by moving your arm through the air for one second.
This calculator gives back the Lorentz factor \(\gamma\). This is a measure of how much things like time, mass, and length need to be adjusted by in coordinate systems moving at relativistic speeds. It’s calculated using this formula, where c is the speed of light and v is the velocity of the reference frame in question:
\[ \gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} \]
For example, at about 75% of c, the masses of objects are around 50% more than they are at rest. I’ve also included the inverse of the Lorentz factor, so you can easily see that time runs at about 2/3 the speed.
There are some interesting things you might note while playing with this. The difference between 1m/s and 1000m/s is almost negligible at low speeds (in the parts per billion) but once you’re at ninety-nine point something percent of the speed of light, the difference can produce big changes in Lorentz factor.
You have to get all the way up to around 14% c before time starts running one percentage point slower (99% of the speed of “normal” time). Once you get there though, it takes only five more percentage points (19% of c) to get to 98%.