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1.10. Modeling with Functions

An important feature of functions is that they can be used to model many scientific phenomena, such as motion, electricity, radioactivity, and population growth, to name a few. In this section we will consider the law of falling bodies, which is modeled very well by simple functions we have studied. Later we will consider more complicated models.

The law of falling bodies is as follows. If an object, such as a stone, is in free fall, i.e. falling without air resistance, then the distance it falls after time t is approximately given by

**(1.10.1)**y(t) = 4.9t^{2}

where t is the time in seconds from when the object is dropped and y is the distance in meters in which it has fallen. Furthermore, the speed at which it is falling after time t is approximately given by

**(1.10.2)**v(t) = 9.8t

where once again, t is the time in seconds from when the object is dropped and v is the speed in meters per second at which it is falling.

**Example 1: **A ball is dropped from a high cliff. How far has it fallen at at what speed is it falling after 3 seconds?

**Solution: **We subtitute t=3 into Equation (1.10.1) to determine how far the ball has fallen. We obtain y(3 )= 4.9 * 3^{2} = 4.9 * 9 = 44.1 meters. To determine how fast the ball is falling, we plug in t = 3 into Equation (1.10.2), obtaining v(3) = 9.8 * 3 = 29.4 meters per second.