Please show all work. Thanks! Suppose that a particle follows the path r (t) = i

Question

Please show all work. Thanks!

Suppose that a particle follows the path r (t) = i sin(3t) + 3cos(3t)j. Give an equation (in the form of a formula involving .r and y set equal to 0) whose solutions consist of the path of the particle. xsin(3t)+3cos(3t)y = 0 (Answer in terms of X and J/.) Determine the velocity vector of the particle when t = pi v(pi) = (3cos(3pi))i-9sin(3pi)j (Answer in terms of t.) Determine the acceleration vector of the particle when t = pi a(pi) = -9sin(3pi)i-27cos(3pi)j (Answer in terms off.)

Explanation / Answer

from above equation as r(t) =isin(3t) +3cos(3t)j

hence x =sin(3t) and y =3cos(3t)

therefore x2+y2/9 =sin2(3t) +cos2(3t)

hence  x2+y2/9 = 1 is the solution of particle path,

now as V(t) =d(r(t)/dt =3cos(3t)i -9sin(3t)j

hence V(pi) =3cos(3pi)i -9sin(3pi)j

also a(t) =d(V(t))/dt

a(t) =-9sin(3t)i-27cos(3t)j

a(pi) =-9sin(3pi)i-27cos(3pi)j

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