Question: Before Galileo discovered that the speed of a falling body with no air resistance is proportional to the time since it was dropped, he mistakenly conjectured that the speed was proportional to the distance it had fallen.
(a) Assume the mistaken conjecture to be true and write an equation relating the distance fallen, D(t), at time t, and its derivative.
(b) Using your answer to part (a) and the correct initial conditions, show that D would have to be equal to 0 for all t, and therefore the conjecture must be wrong.