Lynn Marecek; MaryAnne Anthony-Smith; Andrea Honeycutt Mathis; Jay Abramson; Sharon North; Amy Baldwin; Alyssa Howell

7.5.3 Distance as a Quadratic Function of Elapsed Time

7.5.3 • Distance as a Quadratic Function of Elapsed Time

Activity

Galileo Galilei, an Italian scientist, and other medieval scholars studied the motion of free-falling objects. The law they discovered can be expressed by the equation $d = 16 t^{2}$ , which gives the distance fallen in feet as a function of time, $t$ , in seconds.

An object is dropped from a height of 576 feet.

1. How many feet does it fall in 0.5 seconds?

It falls 4 feet in 0.5 seconds because $16 (0.5)^{2} = 4$ .

2. To find out where the object is the first few seconds after it was dropped, Elena and Diego created different tables.

Elena’s table:

Time (Seconds)	Distance Fallen (Feet)
0	0
1	16
2	64
3
4
$t$

Diego’s table:

Time (Seconds)	Distance from the Ground (Feet)
0	576
1	560
2	512
3
4
t

a. How are the two tables alike?

Compare your answer:

They both have time in seconds and distance in feet as the two quantities being measured. They both show 0–4 seconds and $t$ seconds for time. They describe the same object falling.

b. How are the two tables different?

Compare your answer:

Elena’s table measures distance fallen, and Diego’s table measures distance from the ground. They record measurements from opposite ends. Elena’s table starts with 0 feet for 0 seconds, and Diego’s starts with 576 feet for 0 seconds.

3. Complete Elena’s table. Be prepared to explain your reasoning.

a. What is the distance the object has fallen 3 seconds after it was dropped?

144. The object has fallen 144 feet 3 seconds after it was dropped.

b. What is the distance the object has fallen 4 seconds after it was dropped?

256. The object has fallen 256 feet 4 seconds after it was dropped.

c. What is the distance the object has fallen $t$ seconds after it was dropped?

Compare your answer:

$16 t^{2}$

4. Complete Diego’s table. Be prepared to explain your reasoning.

a. What is the object’s distance from the ground 3 seconds after it was dropped?

432. The object is 432 feet from the ground 3 seconds after it was dropped.

b. What is the object’s distance from the ground 4 seconds after it was dropped?

320. The object is 320 feet from the ground 4 seconds after it was dropped.

c. What is the object’s distance from the ground $t$ seconds after it was dropped?

Compare your answer:

$576 - 16 t^{2}$

Video: Distance as a Quadratic Function of Elapsed Time

Watch the following video to learn more about quadratic functions.

Distance as a Quadratic Function of Elapsed Time

Self Check

A small ball is dropped from a 350-foot-tall building. Which equation could represent the ball’s height,

h(t)

, in feet, relative to the ground, as a function of time,

t

, in seconds?

$h(t)=\frac{350}{16t^2}$
$h(t)=350+16t^2$
$h(t)=350-16t^2$
$h(t)=350-16t$

Additional Resources

Height as a Quadratic Function

The general formula for a free falling object is:

$h (t) = - 16 t^{2} + v_{0} t + h_{0}$

Let’s look at the meaning of the coefficients in the formula:

In the formula, $- 16$ is the constant that has been determined by a combination of Newton’s Laws and Earth’s weight and is used with the units feet/second $^{2}$ .

$v_{0}$ is the vertical speed of the object. If the object is simply dropped, the vertical speed is 0. If the object were thrown upward, this value would be positive, and if the object were thrown down, this value would be negative.

$h_{0}$ is the initial height of the object.

A man drops his keys from a restaurant at the top of a 196-foot-tall building. If the keys are free falling, will they reach the ground within 3 seconds?

Step 1 - Write an equation for the height at a given time, $t$ .

$h (t) = 196 - 16 t^{2}$

Step 2 - Substitute $t = 3$ into the function.

$h (3) = 196 - 16 (3)^{2}$

Step 3 - Evaluate.

$h (3) = 196 - 144 = 52$

At $t = 3$ seconds, the keys will still have 52 feet to travel to hit the ground.

Try it

Height as a Quadratic Function

Will the keys in the situation above reach the ground after 5 seconds?

Here is how to determine if the keys reach the ground after 5 seconds:

Step 1 - Write an equation for the height at a given time, $t$ . $h (t) = 196 - 16 t^{2}$

Step 2 - Substitute $t = 5$ into the function. $h (5) = 196 - 16 (5)^{2}$

Step 3 - Evaluate. $h (5) = 196 - 400 = - 204$

The keys reach the ground before $t = 5$ seconds since substituting in 5 creates a negative height, which implies they have fallen below ground level and does not make sense for this scenario.