Question

Suppose that the position of one particle at time t is given by Xi= 3 sint Yi = 2 cost 0 < t < 211 and the position of a second particle is given by X2 = -3+ cost y2 = 1+ sin t 0<t<21 (a)Graph the paths of both particles. How many points of ntersection are there? (b) Are any of these points of intersection collision points? In other words, are the particles ever at the same place at the same time? If so, find the collision points. (c) Describe what happens if the path of the second particle is given by X2=3 +cost y2=1+ sint 0 < t < 210

Answer 1

Given position of particle one at time t as I1 = 3sin(t), Y1 = 2cos(t) 0<t < 27 .. sinº(t) + cos? (t) = ?)2 +*2 as sinº(t) + cos²(t) = 1 z yi 1 9 4 This equation represents equation of the ellipse. Position of particle two at time tas 12 = -3+ cos(t), y2 = 1+ sin(t) 0<t <21 sin(t) = y2 - 1 cos(t) = 22 +3 :. sin?(t) + cos²(t) = (y2 - 1)2 + (12+3)2 = (y2 – 1)2 + (12 + 3)2 = 1 This equation represents equation of the circle with unit radius. 0 X We can see path taken by particle one and particle two in the above graph. a. When two paths intersect, at the point of intersection Ti = 12 and Y1 = y2. But the time o intersection need not be same. (-2.097, 1.43) 0 X (-3,0) From graph we can see that there are two points of intersection at (-3,0) and (-2.097, 1.43). :. Number of Intersection points=2.

b. The collision points are the points of intersection when I1 = 12 and Y1 = y2 at same time. This means that not only the path intersects but both particles meet. ..11 =12 =3sin(t) = -3+cos(t) + (1) Y1 = 42 = 2cos(t) = 1 + sin(t) + (2) Multipy equation (1) by 2. Add it to equation (2) → 6sin(t) + 2cos(t) = -6 +2cos(t) +1+ sin(t) = 5sin(t) = -5 = sin(t) = -1 =t= sin (-1) 371 t= 2 3л ... At 2 , particles collide. 37 Substitute in Ii and Y1 to get the collision point at time 3л. 2. 11 = 3sind 11 =-3= 12 3л. Y1 2cos y1 = 0 = y2 The collision point is (-3,0). : Number of collision point =1.
c.

Now position of particle two at time tas 22 = 3 + cos(t), y2 = 1 + sin(t) 0<t < 27 sin(t) = y2-1 cos(t) = 12 - 3 (42 – 1)2 + (02 - 3)2 = 1 This equation represents equation of the circle with unit radius. (2.097, 1.43) 0 X (3, 0) From graph we see that there are two points of intersection at (3,0) and (2.097, 1.43). :. Number of intersection points=2.