        Solutions to Exercise 23

 back to exercise motion: TOC for Knowledge Concepts, Exercises, and Solutions

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The Cassini spacecraft is traveling (30,000 m/s)/(300,000,000 m/s) = 0.0001 times
the speed of light.

(a) Time dilation is at work here.  t' = t * sqrt(1 - (v/c)^2), so

3 yr = t * sqrt(1 - (0.0001)^2) = 2.999999985 yr

This means Cassini's onboard clock will have recorded
0.000000015 * 86400 * 365.25 = 0.47 second less time than the clock on Earth.

(b) Length contraction is at work here.  l' = l / sqrt(1 - (v/c)^2), so

3,000,000 km = l / sqrt(1 - (0.0001)^2) ;  thus,

l = 3,000,000 * 0.999999995 = 2,999,999.985 km

This means Cassini sees the distance to be 15 meters less than 3,000,000 km,
or about 50 feet of contraction over a length of 1.86 million miles.

Clearly, even our speediest spacecraft barely notice the effects of special
relativity in their motion.  But microscopic particles often travel at speeds far
exceeding that of our fastest vehicles; for them, special relativity has a major
effect.

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