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

v_{o} represents initial velocity

s_{o} 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/sec
^{2}) = 16 ft/sec
^{2}. Since
g, or the
acceleration due to gravity, is being measured in ft/sec
^{2}, we must also measure
s(t),
v_{o}, and
s_{o} 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.