Draw the acceleration-time graph given a parabolic position-time graph| University Physics

The main graph shows position versus time for a certain object. Which of the graphs below correctly depicts the object’s acceleration?




The given position vs time graph is a parabola, a second degree curve. The slope of the curve is positive from time 0 to 10 seconds, then negative from 10 to 20 seconds. The derivative of a second degree function is a first degree function, or a linear function. This means that the velocity-time graph of this is a straight line coming from positive going negative. The slope of the velocity-time graph is negative constant because it is falling to the right. The slope is the constant acceleration of the given problem. Therefore, the acceleration-time graph of this is a constant negative, a horizontal line in the negative y. The only obvious answer is graph A.