Question

Why are curves that are allowed and common on position-time graphs and not
often found on velocity-time graphs presented?

Answer 1

The reason why curves common on position-time graphs are not often found on velocity-time graphs is that the systems of coordinates are different. Velocity is the first derivative of position, which means that they have different power, and thus, shape. Assume that the position is defined by the following function:

$$x(t)=kt^2.$$

The velocity that corresponds to this function is

$$v(t)=x'(t)=2kt^{2-1}=2kt.$$

As we see, the position is a quadratic function of time (the curve is a parabola), whereas the velocity is a linear function of time (the "curve" is a line).