When does it pay to have a formula to calculate the value of a particular quantity? A formula’s utility depends on relative frequency and relative simplicity, that is, how often we need to determine the value of the quantity isolated in the formula and how simple the formula is by comparison with the step-by-step algorithm for finding the designated quantity. In a number of postings to this blog, I have developed formulas with no concern for the formula’s eventual utility and I have received no complaints, so here are three more.
If a ball dropped from height h always returns to a height that is r times h, where 0 < r < 1, what formula can we use to determine the total distance D traveled by the ball when it finally comes to rest?
Table 1 shows the distance traveled by the ball in each move. In each downward move, except for the first one, the ball travels the same distance it travels in the preceding upward move.
Table 1: Up r h r ²h r ³h
. Down h r h r ²h r ³h
The total distance traveled by the ball is, therefore, h + 2 (r h + r ²h + r ³h + … ), where the parenthetical expression is clearly the sum of an infinite geometric sequence. This sum is determined by the formula, , where is the first term of the sequence and r is the common ratio of any term to its predecessor. Hence,
D + , so D
Continuing with the same variables, what is the total time T it takes for the ball to come to rest?
The distance s traveled by a falling object in time t is given by the formula , where g is the acceleration of gravity. This is the same distance traveled by an object moving upward with an initial velocity equal to the terminal velocity of the falling object. Solving for t in terms of s and g yields . Using the distances in Table 1, Table 2 shows the time taken for each move.
Table 2 Up
The total time it takes for the ball to come to rest is, therefore, + 2 ( + + + …) , where the parenthetical expression is the sum of an infinite geometrical sequence in which the first term is and the common ratio is . Using the formula for the sum of an infinite geometrical sequence, we get
, so . If g = 32 ft/sec² is adequate for a particular context, this can be simplified to .
What is the mean velocity V of the ball over the total distance traveled?
Using , where is mean velocity, is distance traveled and is elapsed time, we get
, so . As in the formula for T, if the aforementioned value for g is used, this can be simplified to .
These three formulas are consistent with the standard d rt formula. Substituting the three values we have derived for d, r and t in the situation under discussion, we get • , which is revealed to be an identity by simplifying the right side of the equation. Readers may determine for themselves the formulas’ utility.