Let s(t) denote the vertical displacement of a body in free fall (downward positive) and v(t) its instantaneous velocity. Answer the following questions without assuming any specific numerical values.
(i) Starting from the definition of instantaneous velocity, show that the slope of the position–time graph at any instant equals the ordinate of the velocity–time graph at the same instant.
(ii) Prove that the area under the velocity–time graph between times t1 and t2 equals the change in displacement s(t2)−s(t1).
(iii) For motion with constant acceleration g and initial velocity u=0, use parts (i) and (ii) to derive the algebraic relation s=21gt2.