Homework Soursepoints a and b move at constant speed in the counterclockwis
points a and b move at constant speed in the counterclockwise direction on the circles x^2+y^2=4 and x^2+y^2=2.25, respectively, where the units are in meters. At t=0 both points are located on the positive x-axis. Point a completes a full revolution every 6 seconds, and b every 15 seconds. When t=24 seconds, the rate of change of the distance between a and b is
Solution
distance between a and b be d
d^2=(x2-x1)^2+(y2-y1)^2
=2.25+4-2(x1x2+y1y2)
d(d^2)/dt=2(x2-x1)(dx2/dt-dx1/dt)+2(y2-y1)(dy2/dt-dy1/dt)
2=2t=2/6 *24=8
1=1t=2/15 *24=3.2
dx2/dt=vsin2=2*2*pi/6 *sin(8*pi)=0
dy2/dt=vcos2=2*2*pi/6=2.094
dx1/dt=1.5*2*pi/15 *sin(3.2*pi)=-0.369
dy1/dt=1.5*2*pi/15 *cos(3.2*pi)=-0.508
d(d^2)/dt=2*(2-1.5*cos(3.2*pi))*(0+0.369)+2*(0-1.5*sin(3.2*pi))*(2.094+0.508)
=6.96
d(d)/dt=0.5*6.96/d=0.5*6.96/sqrt((2-1.5*cos(3.2*pi))^2+(0-1.5*sin(3.2*pi))^2)=1.044
