A particle travels counter-clockwise in circular motion in the x-y plane around a circle of radius b centered at the origin. The speed of the particle is given by $v = v_1(1 - \cos(ct))$. The constants b, c, and $v_1$ are positive. The particle starts at t=0 on the positive x-axis and travels only one revolution around the circle during the time interval $0 \le t \le \frac{2\pi}{c}$. For parts (i-iv), give the following answers in terms of t, b, and c in simplified form using plane polar coordinates during this specified time interval.
(i) r(t), v(t),a(t), and $\Theta$(t)
(ii) The position(s) when the speed is a maximum,
(iii) The position(s) when the speed is equal to $v_1$.
(iv) The time(s) when the position vector is perpendicular to the velocity.
(v) During: $0 \le t \le \frac{2\pi}{c}$, are there any times when the velocity makes a 45° angle with the acceleration? Explain how you know this, and write a simplified equation that could be used to determine the time.
