# Speed, Position and Arc Length

Two identical bugs start moving at the same time on a

flat table, each at the same constant speed of 20

cm/min. Assume that initially (i.e. at time t = 0)

bug 1 is located at point (1, 1) and bug 2 is located

at the point (-1, 1). Assume that units in the

xy-plane are measured in meters and time is measured

in minutes. Further assume that the paths of bug 1

and 2 are given respectively by

C1 : x = a*e-alpha*cos (alpha), y = a*e-alpha*sin

(alpha), pi/4 <= alpha < infinity

C2 : x = a*e-beta*cos (beta), y = a*e-beta*sin

(beta), 3(pi)/4 <= beta < infinity

where a and b are constants.

1. Find the exact values of the constants a and b.

2. Find the arc-length of C1 and C2. Use this

information to show that both bugs reach the origin at

the same time To and find the exact value of To.

3. Find the relationship between the parameter alpha

and time t. What is the relationship between beta and

time t?

4. Find the exact distance between bug 1 and bug 2 at

any time t with 0 <= t < To. Use this information to

conclude that bug 1 never captures bug 2 before t =

To.

5. Find the time at which bug 1 is 2 cm from bug 2.

6. How many times does bug 1 wind around the origin

during the time interval 0 <= t <= 0.9999To? Discuss

the motion of bug 1 on the time interval 0.9999To < t

<= To.

Answer to your queries :

C2 : x = b*(e^(-beta))*cos (beta), y =

>> b*(e^(-beta))*sin(beta)

C1 : x= a*(e^(-alpha))*cos(alpha),

y=a*(e^(-alpha))*sin(alpha)

#### Solution Summary

Speed, Position and Arc Length of two moving objects are investigated.