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Word Problems: Quadratic Max/Min Application - Projectiles
Quadratic Equations are often used to find maximums and minimums for problems involving projectile motion. For example, you would use a quadratic equation to determine how many seconds would be needed for a ball to reach its maximum height when it was thrown directly upward with an initial velocity of 96 feet per second from a cliff looming 200 feet above a beach.
In order to solve quadratic equations involving maximums and minimums for projectile motion, it is necessary to
Let’s solve the example of a quadratic equation involving maximums and minimums for projectile motion given above: 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 its maximum height? And, what is the maximum height?
To analyze our problems, we will be using a formula for a freely falling body in which we can ignore any effects of air resistance.
  • s(t) represents the projectile's instantaneous height at any time t
  • vo represents initial velocity
  • so represents the initial height from which the projectile is released
  • t represents time in seconds after the projectile is released
In this formula, -16 is a constant is based on the gravitational force of the earth and represents ½ g = ½(-32 ft/sec2) = -16 ft/sec2. Since g, or the acceleration due to gravity, is being measured in ft/sec2, we must also measure s(t), vo, and so in terms of feet and seconds.
Let's begin by substituting known values for variables in the formula:

Since the formula represents a parabola, we must find the vertex of the parabola to find the time it takes for the ball to reach its maximum height as well as the maximum height (called the apex) . Using the vertex formula:

Substituting into the projectile motion formula we have:

Therefore, if a ball is thrown directly upward from an initial height of 200 feet with an initial velocity of 96 feet per second, after 3 seconds it will reach a maximum height of 344 feet.

Let's Practice

Question #1
AudioSome fireworks are fired vertically into the air from the ground at an initial velocity of 80 feet per second. Find the highest point reached by the projectile just as it explodes.

Question #2
AudioA ball is thrown vertically upward with an initial velocity of 48 feet per second. If the ball started from a height of 8 feet off the ground, determine the time it will take for the ball to hit the ground.

Try These
Question #1
AudioA pistol is accidentally discharged vertically upward at a height of 3 feet above the ground. If the bullet has an initial muzzle velocity of 200 feet per second, what maximum height will it reach before it starts to fall to the ground?

Question #2
AudioAn over zealous golfer hits a flop shot with a sand wedge to get out of the corner of a sand trip with an initial velocity of 45 feet per second. What is the maximum height that the golf ball will reach?

For problems of this type you must know how to use the projectile height formula and the vertex formula for a parabola. You must be able to correctly substitute in the correct values (measured in feet and seconds) for each variable and be able to use the Pythagorean Theorem when needed.

D Saye

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