When The Shuttle Bus Comes To A Sudden Stop To Avoid Hitting A Dog, It Accelerates Uniformly At -5.1 M/s^2 As It Slows From 8.3m/s To 0m/s. How Can I Find The Time Interval Of Acceleration For The Bus?

Unless I'm missing something here, my understanding of your question involves a bus attempting an emergency stop to avoid hitting a dog. If this is correct so far, the object is to bring the bus to the most immediate and fastest stop humanly possible. So my first inquiry would be to try and determine the negative derivative of the buses deceleration in its rate of change in velocity over time, divided by the derivative rate of change in its velocity with respect to its change rate relative to time. Keeping in mind, that an objects velocity is the derivative (or rate of change) of its speed, while at the same time, the acceleration of an object is, similarly the derivative of that object's velocity. In your question, you stated that the bus was attempting a sudden stop while uniformly accelerating. My calculus may be a little rusty, or else something might have simply floated over my head. If that is the case, I would appreciate your filling in the Swiss Cheese Holes in my gray matter, if there is any left.