| |||||||

Solutions to Exercise 24
(a) Since the speed of light is 300000 km/s, in its lifetime the muon could be expected to travel 300000 km/s * 0.0001 s = 30 km. This would leave it 70 km away from the surface when it decays. (b) The time experienced by the muon, t', is related to the time experienced by an observer watching the muon, t, by the equation t' = t * sqrt(1 - v^2/c^2) where v is the velocity of the muon. Thus, as the muon experiences 0.0001 second while traveling at 98% the speed of light (or 0.98 * c), the observer would experience t = 0.0001 s / sqrt(1 - (0.98 * c)^2/c^2 ) = 0.0001 s / sqrt(0.0396) = 0.00050 s (c) As the Earth-bound observer watches, then, the muon actually can go r = v*t = 0.98 * 300000 km/s * 0.0050 s = 148 km So time dilation due to special relativity means the muon will make it to Earth's surface, 100 km below, before it decays! (d) The distance to Earth as seen by the muon, l', is related to the distance as seen by an Earth-bound observer, l, by the equation l' = l * sqrt(1 - v^2/c^2) where v is the velocity of the muon. Thus, as the muon travels at 0.98 * c, it experiences a distance of l' = 100 km * sqrt(1 - (0.98 * c)^2/c^2) = 20. km (e) The muon travels this 20 km distance in t = r/v = 20. km/(0.98 * 300000 km/s) = 0.000068 s which is less than its lifetime of 0.0001 second. So length contraction due to special relativity means the muon will make it to Earth's surface a full 0.000032 second before it decays! Now you may ask, "which effect is actually working here, time dilation or length contraction?" Well, both are going on. These are just two different ways of looking at exactly the same series of events. We are used to seeing things transpire as our rate of spatial motion changes - for example, our brains easily compensate when we watch a pedestrian on the sidewalk as we ride by in a car, even as the car speeds up and slows down. But we are NOT used to seeing things transpire as the rate of temporal motion changes - our cars never approach relativistic speeds, so time never appears to speed up or slow down. But Einstein showed, using special relativity, motion through time is just as variable as motion through space. So although it takes some effort for our brains to compensate and grasp this concept of variable speeds in time, it's worth a try because that's how the universe works! |