 Site Navigation                            Word Lesson: 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 which represents one-half of the acceleration due to gravity on the "surface" of the earth (-32 ft/sec2).
In physics, the variable g represents the gravitational field strength (g = F/m) on the surface of the earth. This property is measured in an unusual unit: lbs/slug where lbs measures the gravitational force between an object and the Earth based on the object's mass, the mass of the Earth, and the distance between them; a slug is the English system's unit for mass. You can see this unit's common use when you say that a massive object acts "sluggishly." Through a series of conversions, the unit lbs/slug can be shown to be equivalent to our unit ft/sec2.
Since this constant is 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:   seconds

Substituting into the projectile motion formula we have:   feet

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. Example Group #1
No audio files were recorded for this set of examples. Some 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. What is your answer?  A 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. What is your answer? Example Group #2
No audio files were recorded for this set of examples. A 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? 628 feet 1,878 feet 20.87 feet 199.33 feet What is your answer?  An 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? 45 feet 13.19 feet 36.64 feet95.26 feet What is your answer? 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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