P16_022.PDF

22. They pass each other at time t ,at x 1 = x 2 = 2 x m where

x 1 = x m cos( ωt + φ 1 )d x 2 = x m cos( ωt + φ 2 ) .

From this, we conclude that cos( ωt + φ 1 )=cos( ωt + φ 2 )= 2 , and therefore that the phases (the

arguments of the cosines) are either both equal to π/ 3oroneis π/ 3 while the other is

π/ 3. Also at

this instant, we have v 1 =

−

v 2

=0where

v 1 =

−

x m ω sin( ωt + φ 1 )d v 2 =

x m ω sin( ωt + φ 2 ) .

sin( ωt + φ 2 ). This leads us to conclude that the phases have opposite

sign. Thus, one phase is π/ 3 and the other phase is

−

π/ 3; the ωt term cancels if we take the phase

diﬀerence, which is seen to be π/ 3

−

(

π/ 3) = 2 π/ 3.

−

This leads to sin( ωt + φ 1 )=

−