Consider a particle that moves in a plane according to the function r(t) = (sin t2, cost2)with an intial position (0,1) at t=0.What is the length of the path required for this particle to return to its intial position?If you were watching a race between two runners, one moving according to x=sin t²and y=cost² and the other according to x=sin(t) and y=cos(t), who would win and whenwould one runner pass the other runner?

To Find : a) What will be the length of the path required for this particle to return to its initial position? b) If you are watching a race between two runners, one moving according to x=sint2 and y=cost2 and other x=sint and y=cost , who would win and when would one runner pass the other runner? Consider the function rt=sint2 , cost2 with the initial position (0, 1) at t=0. The path described by these equations is circular. Now we have to prove this result by squaring both x and y . x2+y2=sin2t2+cos2t2⇒x2+y2=1 Hence, the path of the particle is by the circle with a radius 1. The motion is not periodic as time increases quadratically. Now, we can compute the time taken by 1) runner to complete its first full lap. Since the sine and cosine functions have a period 2π then we must find at which time the argument equals 2π(first cycle) T2=2π⟹T=2π The particle or runner 1) returns to its original position after T=2π And the length of the circular path with a radius r=1 is calculated as follows : L=2πr⇒L=2π, Now consider a second runner 2) moving according to the parametric equation x=sint , y=cost which also follows a circular path given by the equation x2+y2=1. This motion is periodic with the periodT=2π. We already know that 1) will win in a full cycle race since the particle in a return to the starting point after 2π whereas particle 2) takes 2π to complete the cycle. But at the start of the race 2) starts faster than 1). To compute the time at which 1) takes over we equate the parametric equations, x1=x2⟹sint=sint2⟹t=t2 . As a result, the takeover occurs at, t=1. After this time 1) will stay ahead for the remainder of the race with the lead growing larger as time passes.