1418CHAPTER 37(b) The equations do not show a dependence on acceleration (or on the direction of thevelocity vector), which suggests that a circular journey (with its constant magnitudecentripetal acceleration) would give the same result (if the speed is the same) as the onedescribed in the problem. A more careful argument can be given to support this, but itshould be admitted that this is a fairly subtle question that has occasionally precipitateddebates among professional physicists.4. Due to the time-dilation effect, the time between initial and final ages for the daughteris longer than the four years experienced by her father:tfdaughter–tidaughter=(4.000 y)whereis the Lorentz factor (Eq. 37-8).LettingTdenote the age of the father, then theconditions of the problem requireTi=tidaughter+25.00 y ,Tf=tfdaughter– 25.00 y.SinceTfTi= 4.000 y, then these three equations combine to give a single conditionfrom whichcan be determined (and consequentlyv):54 = 4= 13.5= 0.9973.5. In the laboratory, it travels a distanced= 0.00105 m =vt, wherev= 0.992candtis thetime measured on the laboratory clocks. We can use Eq. 37-7 to relatetto the properlifetime of the particlet0:22002110.9920.9921/tvdtttccv cwhich yieldst0= 4.4610–13s = 0.446 ps.6. From the value oftin the graph when= 0, we infer thantoin Eq. 37-9 is 8.0 s.Thus, that equation (which describes the curve in Fig. 37-22) becomes0228.0 s1( / )1ttv c .If we set= 0.96 in this expression, we obtain approximately 29 s fort.7. We solve the time dilation equation for the time elapsed (as measured by Earthobservers):tt0210 9990( .)