| |||||||

Solutions to Exercise 23
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. |