A Time-Distance Quadratic Model

7.5.2 A Time-Distance Quadratic Model

7.5.2 • A Time-Distance Quadratic Model

Activity

A rock is dropped from the top floor of a 500-foot-tall building. A camera captures the distance the rock traveled, in feet, after each second.

An image of a rock dropped from the top floor of a 500-foot tall building. After 0 seconds, the rock fell 0 feet. After 1 second, the rock fell 16 feet. After 2 seconds, the rock fell 64 feet. After 3 seconds, the rock fell 144 feet. After 4 seconds, the rock fell 256 feet. After 5 seconds, the rock fell 400 feet.

1. How many feet will the rock have fallen after 6 seconds? Be prepared to show your reasoning.

Compare your answer:

The rock will have fallen 500 feet after 6 seconds. The distance fallen increased with each additional second. The rock fell 144 feet between 4 and 5 seconds, so between 5 - 6 seconds, it would fall more than 144 feet. After 5 seconds of falling, there is less than 144 feet to go before the rock hits the ground.

2. Jada noticed that the distances fallen are all multiples of 16. She wrote down: $16 = 16 \cdot 1$ $64 = 16 \cdot 4$ $144 = 16 \cdot 9$ $256 = 16 \cdot 16$ $400 = 16 \cdot 25$

Then, she noticed that 1, 4, 9, 16, and 25 are $1^{2}$ , $2^{2}$ , $3^{2}$ , $4^{2}$ , $5^{2}$ .

a. Use Jada’s observations to predict the distance fallen in feet after 7 seconds. (Assume the building is tall enough that an object dropped from the top of it will continue falling for at least 7 seconds.) Be prepared to explain your reasoning.

Compare your answer:

784

b. Write an equation for the function, with $d$ representing the distance dropped after $t$ seconds.

Compare your answer:

$d = 16 \cdot t^{2}$

Self Check

An object is dropped from a building and falls so that the distance in feet it falls,

d(t)

, is found by

d(t)=16t^2

, after

t

seconds. How many feet has it fallen after 10 seconds?

2,560
1,600
320
160

Additional Resources

Finding Distance Fallen

The Empire State Building is 1250 ft high (excluding the lightning rod). A visitor drops a penny from the top, and it falls straight down. How far has it fallen after 5 seconds?

Recall the previous activity’s formula $h (t) = 16 t^{2}$ .

Substitute $t = 5$ .

$h (5) = 16 (5)^{2} = 16 \cdot 25 = 400$ feet.

Try it

Finding Distance Fallen

Will the penny dropped from the Empire State Building reach the ground after 9 seconds? Be prepared to show your reasoning.

Here is how to determine if the penny will hit the ground by 9 seconds:

Substitute $t = 9$ into $h (t) = 16 t^{2}$ .

$h (9) = 16 (9)^{2} = 16 \cdot 81 = 1, 296$

After 9 seconds, the penny could travel 1296 feet. The Empire State Building is 1250 feet tall, so the penny will have hit the ground by 9 seconds.