Question

1. Suppose that the position of one particle at time t is given by x1= 3 sint yı = 2 cost 0<1<21 and the position of a second particle is given by x2 = -3+ cost y2 = 1+ sin 1 0<t<21 (a)Graph the paths of both particles. How many points of intersection 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 +cos 1 y2=l+ sin 1 0<1<21

Answer 1

Here, we are given the position of the particles, Particle I and Particle II.

We first of all eliminate the parameter t

2 0 Posetion of one particle at time t: = 3 sint, y, - 2 costi Ost <2t here, h - sinti sint; y, cost by tu = sent tcost =) sin²t+cost is an equation of elupse 3 2 And the position of sucond particle is :- Hz = -3+ lost; Y2 = 1 tsent oct<275 (4+3) = cos? t ; (T-13 =>(2₂+3% + (Y2-13 = 1 115 is an equation of curce = sunt 2 2 2 Now for ferst particle, at t=0,[se,= 0, y = 2 And for second partull at t=0, [22= -2 : Y = 1

2 0 Posetion of one particle at time t: = 3 sint, y, - 2 costi Ost <2t here, h - sinti sint; y, cost by tu = sent tcost =) sin²t+cost is an equation of elupse 3 2 And the position of sucond particle is :- Hz = -3+ lost; Y2 = 1 tsent oct<275 (4+3) = cos? t ; (T-13 =>(2₂+3% + (Y2-13 = 1 115 is an equation of curce = sunt 2 2 2 Now for ferst particle, at t=0,[se,= 0, y = 2 And for second partull at t=0, [22= -2 : Y = 1

(a)

After elimination, we get to know that the particle I is moving on elliptical path and particle II is moving on circular path as shown in the figure:

@graph of the path of the parterly, Yz bartice I 2 partile I А ( R 22 . ti -31B -2

Here, t is the angle subtended by the particle.

For partiilio tis the angle, the parterle makes with the positive Y-asust For partuile t is the angle, the partite makes with postlerce Hy-Cesus.

From the figure, it is clear that there are two point of intersection of particles I and II e.g, Points A and B

(b)

Coordinates of the posetion Partule IT Particle time TV2 ( ر2-) (2ر3) (ا را- (5ر3 ) ( ر2 - (2 ره) هر 3) (2 -,ه) رہ ر3) TC 31/2 (2 را ج2

Coordinates of the posetion Partule IT Particle time TV2 ( ر2-) (2ر3) (ا را- (5ر3 ) ( ر2 - (2 ره) هر 3) (2 -,ه) رہ ر3) TC 31/2 (2 را ج2

From the table and figure, there is only one collision point of the two particles, i.e, there is only one time, t= 3π/2 , such that the particles are at the same place.

The collision point is B(-3,0)

(c)

If path of the second particle is changed, the two particles never collide at any point of time as shown in the table and figure below:

If the path of the second particle is: x2 = 3+Lost : Y 2 = 1 tsent; ost<em => (42-3)? +42-1)=1 is a urcle. Nowo 2 paretice I 2 poetice II 3 -2 coordinates of poselion time partale I (0,2) 2 (3,0) (0, 2) 3/, (-3,0) 0,22 pevetele II (4,1 (3,2) (2,1) (3.0 (431) 2