If *h* is the height measured in feet, *t* is the number of seconds the object has fallen and *s* is the initial speed (in ft/sec), then the model for height of a falling object is

*h = –*16*t*^{2} + *st*.

The “–16*t*^{2}” term comes from the acceleration due to gravity. If the value of *h* is in meters and *s* in meters/sec, the equation becomes

** h = –5t^{2} + st**.

**Example 1:**

A ball is thrown vertically upward with an initial speed of 80 ft/sec. How high will the ball be after 3 seconds?

* t* = 3 and *s* = 80 ft/sec

So, *h* = –16(3)^{2} + 80(3)

= –144 + 240

= 96 feet

**Example 2:**

An object is dropped from a height of 120 feet. Assuming that there is no air resistance, how long does it takes to reach the ground?

If *h* is measured in feet, *t* is the number of seconds the object had fallen, and *s* is the initial height from which the object was dropped, then the model for the height of falling object is:

Substitute 0 for *h* and 120 for *s* in the model.

Solve the equation for *t*.

Taking the square root:

Since *t* represents time, it cannot be negative.

Therefore, the object will reach the ground in about 2.74 seconds.

These equations are simplified. They ignore air resistance and the gravitational constant is approximate. Also, this model only works for the surface of the Earth (at sea level). The model on other planets will be different because their gravity is different. For example, on the surface of the moon, with *h* in meters and *s* in m/sec, the falling object model is *h* = –0.8*t*^{2} + *st*.