A particle travels in a straight line from $A$ to $B$ in $20{\text{ }}s$. Its acceleration $t$ seconds after leaving $A$ is $\alpha m{\text{ }}{s^{ - 2}}$, where $\alpha = \frac{3}{{160}}{t^2} - \frac{1}{{800}}{t^3}$. It is given that the particle comes to rest at $B$.
a) Show that the initial speed of the particle is zero.
b) Find the maximum speed of the particle.
c) Find the distance $AB$.
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