144 CHAPTER 1 Equations and Inequalities Height of a Projected Object If air resistance is neglected, the height s (in feet) of an object projected directly upward from an initial height of s0 feet, with initial velocity v0 feet per second, is given by the following equation. s = −16 t2 +v 0 t +s0 Here t represents the number of seconds after the object is projected. The coefficient of t2, -16, is a constant based on the gravitational force of Earth. This constant varies on other surfaces, such as the moon and other planets. Galileo Galilei (1564–1642) According to legend, Galileo dropped objects of different weights from the Leaning Tower of Pisa to disprove the Aristotelian view that heavier objects fall faster than lighter objects. He developed the formula d =16t2 for freely falling objects, where d is the distance in feet that an object falls (neglecting air resistance) in t seconds, regardless of weight. EXAMPLE 3 Solving a Problem Involving Projectile Height If a projectile is launched vertically upward from the ground with an initial velocity of 100 ft per sec, then neglecting air resistance, its height s (in feet) above the ground t seconds after projection is given by s = -16t2 + 100t. (a) After how many seconds, to the nearest hundredth, will the projectile be 50 ft above the ground? (b) How long will it take for the projectile to return to the ground? SOLUTION (a) We must find value(s) of t so that height s is 50 ft. s = -16t2 + 100t 50 = -16t2 + 100t Let s = 50. 0 = -16t2 + 100t - 50 Standard form 0 = 8t2 - 50t + 25 Divide by -2. t = -b {2b2 - 4ac 2a Quadratic formula t = -1-502 {21-5022 - 41821252 2182 Substitute a = 8, b = -50, and c = 25. t = 50 {21700 16 Simplify. t ≈0.55 or t ≈5.70 Use a calculator. Both solutions are acceptable. The projectile reaches 50 ft twice—once on its way up (after 0.55 sec) and once on its way down (after 5.70 sec). (b) When the projectile returns to the ground, the height s will be 0 ft. s = -16t2 + 100t 0 = -16t2 + 100t Let s = 0. 0 = -4t14t - 252 Factor. -4t = 0 or 4t - 25 = 0 Zero-factor property t = 0 or t = 6.25 Solve each equation. The first solution, 0, represents the time at which the projectile was on the ground prior to being launched, so it does not answer the question. The projectile will return to the ground 6.25 sec after it is launched. S Now Try Exercise 47. Substitute carefully.
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