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,

*v*_{o} = 96 ft/sec, and

*s*_{o} = 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*.

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.