Question $2.$ The position vector $r(t)=x(t)e_1+y(t)e_2+z(t)e_3$ of a roller coaster vehicle at Six Flags New England, Massachusetts of mass $m$ denotes the shape of a modified cylindrical helix with components $x(t)=rcos,y(t)=rsin,z(t)=h(t-t^3)$ where $(t)=t,r$ is any positive constant and height $h$ is also a positive constant. Determine the following quantities:

$($i$)$ the unit tangent vector $e_t$ and binormal unit vector $e_b$ at a point $P$ along the path of motion followed by the roller coaster vehicle.

$($ii$)$ the curvature $$ and radius of curvature $$

$($iii$)$ the speed $v=||v||=\sqrt[\phantom{2}]{v*v}$ and the magnitude of the force $||F||=\sqrt[\phantom{2}]{F*F}$

$($iv$)$ the Curl grad$F,$ and if $F$ is not a conservative force, what should be done to make it conservative. Note that the del grad and force $F$ in the three dimensional Cartesian coordinate system are defined as

grad$=\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{x}}{e}_1+\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{y}}{e}_2+\frac{\text{d}\text{e}\text{l}}{\text{d}\text{e}\text{l}\text{z}}{e}_3,F=F_x{e}_1+F_y{e}_2+F_z{e}_3,$

and the notations $F_x,F_y$ and $F_z$ are the scalar components of the force vector $F.$

$($v$)$ describe the differences between conservative and nonconservative forces

$($vi$)$ the angular momentum $h_O=rp$ and the rate of change of angular momentum $\frac{d{h}_O}{dt}=\frac{d}{dt}(rp),$ where the linear momentum $p=mv$

$($vii$)$ why is the total mechanical energy $E=T+U$ a scalar quantity and moment of a force or torque ${}_O=rF$ a vector quantity