Question

2. In R3 you are given the points P(0,0, 10) and Q(42, 70, 4), and R(42, 70,-4) (a) Find the equation of the linear motion which travels through P at time 0 and through R at time 7 (b) Describe the motion of the particle which travels via linear motion, passing through P at time 0, then bounces off of the ry-plane and continues via a linear motion, until it passes though Q at time 14. (Draw a sketch of the situation, and assume that the bounce off of the ry-plane behaves like the reflection of a light ray in a mirror. Then construct the finadian vides for cerh lastane at time t.)

Answer 1

The linear equation for the position P(t) ()(t)) is calculated below. x(t) 0+42-0 7-0 6t 70-0 7-0 10t And, こ(7)=10+(-4-10 7-0 -10-2t Hence, the position of the particle is P(t) (6r,10r, 10-2)

The diagram for the question is drawn below P(0,0,1 R(42,70,-4) (0,0,10 0

In triangles POM and RSM. PM OM OP 10 RM SM SR 4 Let t be the time taken by the particle in travelling from P to M and t, be the time taken by particle in travelling from M to R. i PM 2 RM Therefore, 1+1+1 t. It is given that t +t-14 Therefore,

1+2 7 14 7 And, ti = 10 s The coordinates of the point M are, 2 9 2 (42-0)5 (70-0)5 7 = (30,50,0)

The parametric equation s of the point P are 30-0 10-0 - (3t, 5t,10-t) 50-0 10-0 The parametric equations of M are 14-10.50+70-50 (30+3r,50+5t,t) The function f(t) is given by, f(t)- 3t,5t,10-t 30+3t, 50+5t,t if 10<t <14 if0<t <10