P16_023.PDF

23. (a) Let

2 cos 2 πt

x 1 = A

be the coordinate as a function of time for particle 1 and

x 2 = A

2 cos 2 πt

T + π

be the coordinate as a function of time for particle 2. Here T is the period. Note that since the

range of the motion is A , the amplitudes are both A/ 2. The arguments of the cosine functions are

in radians. Particle 1 is at one end of its path ( x 1 = A/ 2) when t = 0. Particle 2 is at A/ 2when

2 πt/T + π/ 6=0or t =

−

x 1 = A

2 cos 2 π

0 . 50s

1 . 5s

−

0 . 250 A

and

2 cos 2 π

x 2 = A

0 . 50s

1 . 5s

+ π

−

0 . 433 A.

Their separation at that time is x 1 −

x 2 =

−

0 . 250 A +0 . 433 A =0 . 183 A .

(b) The velocities of the particles are given by

v 1 = d x 1

d t = πA

T sin 2 πt

and

T sin 2 πt

v 2 = d x 2

d t = πA

T + π

We evaluate these expressions for t =0 . 50s and ﬁnd they are both negative-valued, indicating that

the particles are moving in the same direction.

T/ 12. That is, particle 1 lags particle 2 by one-twelfth a period. We want

the coordinates of the particles 0 . 50s later; that is, at t =0 . 50s,