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Several 10s km When viewed from the perspective of a muon at rest, the distance to the surface of Earth is several 10s km. When viewed from the per- spective of a muon traveling at nearlight speed, the distance to the surface of Earth contracts to less than 0.6 km. Earth Upper Atmosphere Figure 3-4: Distance contracts as well as time. Contracts to 0.6 km To the muon, Earth appears to be approaching at a velocity close to the speed of light. Mass When Moving Now let's consider how the mass of an object is related to its velocity according to relativity. Let's start by reviewing the laws of motion. Before relativity was understood, the Galilean transformation and Newton's law of motion were used to describe motion. Galilean Transformation The Galilean transformation describes the relationship between coordinate systems moving at velocity v: x = x - vt and t = t where x¢ and t¢ represent position and time, respectively, in one system and x, v, and t rep- resent position, velocity, and time, respectively, in the other system. Newton's Second Law of Motion Newton's second law of motion is represented as follows: f = ma = m d 2 x , dt 2 where f represents force, m represents mass, a represents acceleration, and acceleration can be considered the second derivative of displacement with respect to time: a = d 2 x dt 2 The Faster an Object Moves, the Shorter and Heavier It Becomes? 109