For each of the following situations, sketch a graph that is a possible description of position as a function of time fora particle that moves along the x axis. At t = 1 s, the particle has (a) zero velocity and positive acceleration; (b) zero velocity and negative acceleration; (c) negative velocity and positive acceleration; (d) negative velocity and negative acceleration. (e) For which of these situations is the speed of the particle increasing at t = 1 s?

(a)
The displacement as a function of time is given below:

It can be seen from the graph above that at t = 1 s, , the slope of the graph is zero and hence accounts for zero velocity. Also, the object has positive acceleration and the displacement must time graph must increase non-linearly thereafter.
(b)
The displacement time graph is shown below:


It can be seen from the figure above that the slope of the graph at t = 1 s is equal to zero and accounts for zero velocity. Also, the object experiences negative acceleration and hence displacement time must decrease non-linearly.
(c)
The displacement time graph is shown below:

It can be seen from the figure above that the slope of the graph is negative at t = 1 s and to account for the positive acceleration, the change of slope must increase with time. This can be accomplished only when the graph represent small changes in displacement over time after t = 1 s.
(d)
The displacement time graph is shown in the figure below:

It can be seen from the figure above that the slope of the graph is negative at t = 1 s and to account for the negative acceleration, the change of slope must increase over time. This can be accomplished if the change in displacement is rapid and increases rapidly with small changes in time.
(e)
The speed of the particle increases in graph for part (a) and part (c) shown in the figure below. The increase in speed also accounts for the positive acceleration of the object.

It can be seen from the graphs above that after t = 1 sec , the displacement is such that change of slope is positive accounting for the positive acceleration.