Problem 9: The first man to study and partially understand motion under constant acceleration was Leonardo da Vinci (1452 – 1519), almost two centuries before calculus was invented. He describes,

in his famous notebooks, the apparatus shown consisting of two vertical boards, hinged together on one side and covered with blotting paper on the inside faces. A leaking water faucet lets drops fall between the boards at equal intervals of time. Using a string mechanism, the boards can be quickly clapped together and the position of the drops on the blotting paper can be inspected. Leonardo da Vinci observed that the distance between consecutive drops increased in a “continuous arithmetic progression."! (a) If the water droplets drip from the faucet at a uniform rate of n drops per second, find the distance x between any two adjacent drops as a function of the time t that the trailing drop has been in motion. Neglect air resistance. Assume that the acceleration due to gravity, g, is constant. (b) Suppose that the vertical boards are 3 ft long. The leaking faucet is at such a height, and therate of water droplets falling is so regulated, that when a drop is just coming out of the faucet,the next or second drop is just at the top of the boards, and the sixth drop is at the bottom ofthe board. Calculate the number of drops leaving the faucet each second. Also determine thedistances between the six drops of water.