Consider a particle which starts from rest and has position . What is the color of the curve which represents its velocity as a function of time?
Different possible graphs of velocity vs time.
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Observe that the velocity of an object is the derivative of it's displacement function. Thus, in this case, as the displacement function is modelled by x ( t ) = t 2 , the velocity would be given by:
v ( t ) = d t d x ( t ) = d t d x 2 = 2 t
Clearly the only graph which is satisfying the function: v ( t ) = 2 t is the red line .