You misunderstand what is meant by "moving in slow motion". The concept your wrestling with states "if an electron moves at the speed of light, then it should appear to be moving at 10 miles an hour". That's a contradiction. If it's moving at near the speed of light, it's moving at near the speed of light. Time dilation affects INTERNAL time. If we look at a spaceship, nothing is obviously different about its speed and trajectory; however, look INSIDE the ship and everyone appears to be moving in slow motion. The only way we see this in a particle is by measuring its lifetime. I believe that one of the proofs for relativity was in explaining how a muon made it into the earth's atmosphere from the sun. We are 8 light-minutes from the sun and its lifetime is 7 minutes meaning it would need to travel faster than light to make it here. That's impossible! However, taking into account its relativistic speed and calculating the time dilation for the muon, it actually was "aging" slower and extended its lifetime several minutes, more than enough to make it into our atmosphere before decaying.
<This message has been edited by Victor S (edited 05 November 2000).>