Taking the square root gives v/c = 0.995. The particle must be within one-half of one percent of the speed of light!
Time dilation applies to biological clocks. A person traveling at a high speed will not age as fast as a person at rest. Of course, the situation must be symmetric. Each person sees the other age slower. This leads to a puzzle known as the twin paradox. Two twins are on Earth. One is an astronaut who goes on a trip at a speed close to c. The other stays on the Earth. From the point of view of the one that stayed on Earth, the astronaut is moving and will not age as fast as the one on Earth. The astronaut will appear younger upon returning. However, the astronaut sees the one on Earth moving away at high speed. Therefore the one on Earth should appear younger. It is alright for two moving observers to see each other age slower. However, we have a problem if we try to bring the twins together - both at rest. We can see which one is really younger and decide which was really moving. This would seem to violate Einstein's postulate. However, if the twins start and end together at rest, then one twin must accelerate to get to very high speeds. That acceleration produces pseudoforces which can be felt by only one twin. This breaks the symmetry of the problem without any logical contradiction. (Remember, a pseudoforce is really an inertial response to an acceleration of the reference frame.)
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