I have done the part a) and have some understanding of what is going on in part b), but stack in further actions
At time t a particle has a velocity v(t) = [2cos(t)]i [4sin(t)cos(t)]j, t 0 and at time t = 0 it is at a point which has a position vector r(t) = 3j
a) Find the position of the particle at time t .
b) Find the position of the particle when it first comes to rest .
c) i). Find the Cartesian equation of the path of the particle. ii). Sketch the path of the particle giving the domain and range
d). Express |v|^2 in terms of cos(t) and without using calculus, find the maximum speed of the particle.
e). Give the time at which the particle is at rest for the second time.
f). (i). Show that the distance d of the particle from the origin at time t is given by d72 = cos^2(2t) + 2cos(2t) + 6
(ii). Find the tirne(s) at which the particle is closest to the origin.
For part b, the question is what is the position. We know that velocity is 0. The solution is given by $\cos t =0$. Take this $t$ value, and plug it into the answer for part a.
I was not able to read accurately the formula for $r(t)$ in part a. You should have two components, one along ${\bf i}$ and one along ${\bf j}$. I think you got the first component correct, not sure about the second one. You get a function that contains $\cos 2t$. For part c, you identify $x=2\cos t$, and $y=A\cos 2t+B$. Note that you can write $\cos 2t= 2\cos^2 t-1$. You can now write $\cos t$ from the first equation, and plug it into the second. That's the equation for the path. Note that $x$ is not defined everywhere. You can get the minimum and maximum in terms of $t$