A particle $P$ moves in a straight line. It starts from rest at $A$ and comes to rest instantaneously at $B$. The velocity of $P$ at time $t$ seconds after leaving $A$ is $vm{\text{ }}{s^{ - 1}}$, where $v = 6{t^2} - k{t^3}$ and $k$ is a constant.
a) Find an expression for the displacement of $P$ from $A$ in terms of $t$ and $k$.
b) Find an expression for $t$ in terms of $k$ when $P$ is at $B$.
Given that the distance $AB$ is $108{\text{ }}m$, find
c) the value of $k$,
d) the maximum value of $v$ when the particle is moving from $A$ towards $B$.
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