The magic cart (part 3)

This problem is a further continuation of this problem . Information from the previous problems is also provided for convenience.

A magic cart is placed at the origin of the complex plane. It has two small motors, built into each side of it, which propel it throughout the plane using thrust force. These motors are programmed to function in a way such that the sum of the vectors of the cart's experienced force, momentum and position on the plane, $z$ , is always equal to a certain function, $f(t)$ . The rate at which this sum changes is proportional, by a constant $k$ , to the sum's value at any given time.

A magic rod, connecting the point $P=-2D$ to the center of mass of the cart, is added to the system. This rod magically changes length depending on the distance of the cart from the origin. The cart returns to the origin and carries out the same journey again, this time with the magic rod. If the moment around the point $P$ after $2\pi$ seconds can be expressed as $M_P=a\sin(\tan^{-1}(b))\sqrt{c}D(D+d\pi e^{f\pi}) \ce{Nm}$ where $a,b,c,d$ and $f$ are integers, calculate $abcdf$ .

Details and Assumptions :

• When the cart begins its journey, it has a velocity of $\begin{pmatrix}-0.25\\0\end{pmatrix}$ .
• The initial sum of the force, momentum, and position is equal to $\begin{pmatrix}-1\\0\end{pmatrix}$ .
• The cart has a mass of $3kg$ and each motor has a mass of $0.5kg$ .
• Effects of gravity and friction can be ignored; the only force on the cart is the thrust force.
• The rod has negligible mass and thickness.
• $D$ is the distance between the origin and the cart after $2\pi$ seconds.
• Positive moment is in the anti-clockwise direction; negative moment is in the clockwise direction.
• $\{a,b,c,d,f\}$ is not necessarily a set of distinct integers.
• $k=\lambda i$ for some $\lambda >0$ .
• $i$ denotes the imaginary unit, $i=\sqrt{-1}$ .