Step 1: Read the velocity rule.
The velocity of the particle is $v = a t - b t^{2}$, where $a$ and $b$ are constants. We must find the time when the acceleration becomes zero.
Step 2: Link acceleration to velocity.
Acceleration is how fast velocity changes with time. In calculus terms it is the derivative of velocity: $\text{acc} = \dfrac{dv}{dt}$.
Step 3: Differentiate each term.
The derivative of $a t$ is $a$. The derivative of $b t^{2}$ is $2 b t$. So the acceleration is $\text{acc} = a - 2 b t$.
Step 4: Apply the condition.
We want the moment when acceleration is zero, so set $a - 2 b t = 0$.
Step 5: Solve for time.
Rearranging gives $2 b t = a$, so $t = \dfrac{a}{2b}$.
Step 6: Check the meaning.
At this time the velocity reaches its peak, because acceleration switching from positive to negative marks the highest speed. \[ \boxed{t = \dfrac{a}{2b}} \]