Main problem

The acceleration function of an object moving in a straight line is given by \[a(t) = \sqrt{t} -2.\]

Determine the velocity and displacement functions, and if possible draw the graphs on software such as GeoGebra or Desmos.

Once you have drawn the graphs, answer the following questions. (A GeoGebra applet of the graphs is available here.)

  1. If, at time \(t = 0\), the velocity and displacement are zero, can you describe, draw or imagine what the journey is like?

    Which direction are you travelling in? Are you speeding up or slowing down?

    To draw or describe the journey, think about a straight line with an origin where the object starts (as the displacement is zero).

    Things change about the journey at the following times: \(t = 4\), \(t= 9\) and \(t = 14\). What are they?

  2. Approximately, for what value of \(c\) is there no negative velocity? (We are using \(c\) to represent the constant in the velocity function and \(d\) for the constant in the displacement function.)

    • What feature of the journey changes when there is no negative velocity?

    • What feature of the displacement graph changes when there is no negative velocity?

  3. The acceleration function is negative for the first \(4\) seconds. What does this mean is happening to the object when

    • \(c = 0\)?
    • \(c = 3\)?
    • \(c = 2\)?
  4. Now let’s think about \(c=0\) and \(d=2\). How would you describe the journey, and how does it differ from the journey in question 1?