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Word Lesson: Quadratic Evaluation at a Point
Although quadratic equations are often used to find maximums and minimums for problems involving projectile motion, they can also be used to evaluate the path of a projectile at various time periods. For example, quadratic equations can be used to determine the height of a projectile at time “t” after the projectile has been released.
In order to work a problem involving quadratic evaluation at a point, it is necessary to
Suppose a ball is thrown directly upward from an initial height of 200 feet with an initial velocity of 96 feet per second. After how many seconds will the ball reach a height of 300 feet?
We will begin by substituting our givens in to the projectile height formula: At time t = 0, vo = 96 ft/sec, and so = 200 feet.

The graph of the equation depicting the path of the ball is as follows:
We want to know what the value of t will be when = 300. To find out, we substitute 300 for , and solve the quadratic equation for t.
subtract 300 from each side of the equation
solve for t using the quadratic formula
For a quadratic in the form , the quadratic formula is stated as

We have obtained two values that represent the time that the ball reaches a height of 300 feet. The first value 1.34 indicates that after 1.34 seconds have passed, the ball is at a height of 300 feet. Then the ball reaches its maximum height and begins to fall back to the ground. After 4.66 seconds it is once again at 300 feet. Then it will continue to fall to the ground. The answer we were seeking is 1.34, the time the ball initially reached 300 feet after it has been thrown.

Example A toy rocket is fired vertically into the air from the ground at an initial velocity of 80 feet per second. Find the time it will take for the rocket to return to ground level.
What is your answer?
Example A rock is thrown vertically upward with an initial velocity of 48 feet per second. If the rock toss started from a balcony 3 feet off the ground, determine the time it will take for the rock to reach its highest point (before it begins its descent to the ground). What is that highest point? At what time will the rock hit the ground?
What is your answer?

Example A ball is tossed from 4 feet above ground. It is released with an upward velocity of 50 feet per second. When will be ball be 40 feet above the ground?
  1. Approximately 2.07 seconds
  2. Approximately 1.05 seconds
  3. Approximately 3.85 seconds
  4. Approximately 3.71 seconds
What is your answer?
Example  A bullet is fired vertically into the air from the ground at an initial velocity of 240 feet per second. When will the bullet return to ground level?
  1. -15 seconds
  2. 0 seconds
  3. 15 seconds
  4. 3.87 seconds
What is your answer?

For problems of this type you must know how to use the projectile height formula for a parabola, the vertex formula for a parabola as well either how to factor or use the quadratic formula to solve for time. Graphing your functions will illustrate the projectile's behavior and allow you to understand the mathematics of your answers.

D Saye

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