Question

A rock is thrown upward at an initial velocity of 16 ft/sec from a height of 96 feet above ground.  How long will it take to hit ground?

Answer 1

The equation you are looking for is: . {{{H = -16t^2 + V[o]t + H[o]}}} . where the variables are defined as follows: . {{{H}}} is the height of the rock at time "t" after it is launched {{{t}}} is the time after launch {{{V[o]}}} is the velocity with which the rock is launched {{{H[o]}}} is the height above the ground from which the rock is launched. . First {{{V[o]}}} is given in the problem +16 ft/sec. The sign of this term is plus because the rock is thrown upward. Next {{{H[o]}}} is given in the problem as +96 ft. The sign of this term is plus because the launch point is above the ground level. Substituting these two values into the equation results in: . {{{H = -16t^2 + 16t + 96}}} . One more thing. When the rock finally hits the ground, its height above the ground will be zero. Therefore, substitute 0 as the value of H to make the equation: . {{{0 = -16t^2 + 16t + 96}}} . From here on, this is just a math exercise of solving a quadratic equation. . Note that 16 is a factor of every term on the right side ... therefore, you can divide both sides of this equation (all terms) by 16 and the result is: . {{{0 = -t^2 + t + 6}}} . To get this in a little more standard form, transpose the equation by switching sides, and then multiply the entire equation by -1. When you do that you get: . {{{t^2 - t - 6 = 0}}} . This can be factored to: . {{{(t - 3)*(t + 2) = 0}}} . Notice that this equation will be true if either of the factors is zero because multiplication by zero on the left side will make the left side equal to the 0 on the right side. . So we can find the two possible answers for t by setting each of the factors equal to zero to get: . {{{t - 3 = 0}}} . and . {{{t + 2 = 0}}} . The first is solved by adding 3 to both sides to get {{{t = 3}}}. The second is solved by subtracting 2 from both sides to get {{{t = -2}}}. This second solution can be ignored because a minus value of time means a time before launch. The answer, therefore, is t = 3 seconds. That means that 3 seconds after the launch of the rock (with an initial upward velocity of 16 ft/sec from a height of 96 ft) the rock will hit the ground. . Hope this helps you to understand how to set up the problem and then how to solve it. .