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# Speed, Position and Arc Length

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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.

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).

https://brainmass.com/math/geometric-shapes/speed-position-arc-length-176596

#### Solution Preview

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Two identical bugs start moving at the same time on a flat table, each at the same constant speed of 20cm/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)
C2 : x = b*(e^(-beta))*cos(beta), y = b*(e^(-beta))*sin(beta)
where a and b are constants.

1. Find the exact values of the constants a and b.
Initially, for bug 1, it is at (1, 1), where
So substituting for to the path equation of x:

Initially, for bug 2, it is at (-1, 1), where
So substituting for to the path equation of x:

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.
If a curve is defined ...

#### Solution Summary

Speed, position and arc length of two moving objects are investigated in the solution.

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