Space is where things happen. Time is when things happen. And sometimes, in order to really look at the universe, you need to take those two concepts and mash them together. In this first lesson of a three-part series on space-time, hilarious hosts Andrew Pontzen and Tom Whyntie go through the basics of space and time individually, and use a flip book to illustrate how we can begin to look at them together.
In reality, Tom's car (sadly) doesn't really go fast enough to demonstrate the effects of Lorentz contraction. How fast do you have to be moving before the effects of Lorentz contraction and time dilation do become "noticeable"? You’ll need the formulae for length contraction and time dilation, which can be found here.
At the heart of this lesson is the experimental fact that light moves at a constant speed – 299,792,458 metres per second. Because it's so incredibly fast, people used to think that light was transmitted between objects instantaneously. It wasn't until 1676 that the first measurement of light's finite speed was made by Danish astronomer Ole Rømer. In this video, Professor Brian Cox talks about this measurement and its implications for the nature of the universe. This video from the University of Nottingham's Sixty Symbols series talks more about the speed of light as a fundamental constant of physics and how it is measured today. Finally, this video from UCLA shows how, these days, the speed of light can be measured using standard college laboratory equipment.
In one of his 1905 "Annus Mirabilis" papers [1905c], Einstein took the constant nature of the speed of light as one of two fundamental postulates of his theory of Special Relativity. The other postulate - that the laws of physics are the same in all inertial frames of reference - is the technical way of saying that the laws of physics should be the same for everyone. Einstein won the 1921 Nobel Prize for Physics "for his services to Theoretical Physics", and the Nobel Prize website features a series of resources on his relativity work, as well as the contributions other scientists made.
[1905c] Einstein, A. "Zur Elektrodynamik bewegter Körper" ("On the electrodynamics of moving bodies"), Annalen der Physik 17 (10): 891–921 (1905)
We mention that experimental confirmation of time dilation can be obtained by observing the behaviour of fundamental particles at particle accelerators like the Large Hadron Collider (LHC). Specifically, we can study the observed lifetime of a particle – how long it exists before it decays into other particles. Fundamental particles have a fixed lifetime – for example, the muon (the heavier cousin of the electron) only exists for about 2.2 microseconds at rest. However, if a muon is moving close to the speed of light we will observe it lasting much longer. From the muon's perspective, it still lasts only 2.2 microseconds. But from our persective, time has been dilated in accordance with Einstein's theory.
An experiment performed in 1967 at CERN's Muon Storage Ring  showed that fast-moving muons (accelerated to an energy of 1.274 GeV) appeared to last for 26 microseconds. Another famous experiment to observe the effects of time dilation on very precise clocks was carried out in October of 1971, when four cesium beam atomic clocks were flown around the world in different directions. The predictions and results are reported in [1972a] and [1972b] respectively, but this website contains a good summary of the experiment as well as some photos of the scientists and equipment involved (including the clocks strapped into a plane seat!).
 Bailey, J. et al. "Is the Special Theory Right or Wrong?: Experimental Verifications of the Theory of Relativity", Nature 217:17-18 (1967)
[1972a] Hafele, J. C. and Keating, R. E. "Around-the-world Atomic Clocks: Predicted Relativistic Time Gains", Science 177(4044):166-168 (1972)
[1972b] Hafele, J. C. and Keating, R. E. "Around-the-world Atomic Clocks: Observed Relativistic Time Gains", Science 177(4044):168-170 (1972)