That if an object falls from rest under the constant acceleration of gravity g, its velocity at time t is given by v(t) = gt and the distance that it has fallen is given by s(t) = 1/2gt².

(a) Show that the average velocity of an object falling from rest under the constant acceleration of gravity g between times t = 0 and t = T is given by gT/2. 

(b) Show that the object in part (a) would have a velocity v(s) = √2gs after traveling a distance s.

(c) In part (a), we found the average velocity over time, as usual. Suppose that for the object in parts (a) and (b) we instead found the average velocity over distance. Using the formula from part (b), show that the average velocity over distance as the object falls between distances s = 0 and s = S is given by 2/3√2gs.

(d) Show that if after time T the object has fallen a distance S, the average velocity over distance from part (c), given in terms of T, is 2gT/3. Notice that this is larger than the average velocity over time from part (a). It can be shown that for motion along a line in one direction, the average velocity with respect to distance is always greater than or equal to the average velocity with respect to time.