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<h4>Activity</h4>
<p>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 \(d=16t^2\), which gives the distance fallen in feet as a
function of time, \(t\), in seconds.</p>
<p>An object is dropped from a height of 576 feet.</p>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="3c8e0724-d5b6-418b-b70d-d2a7d974b821" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q1-->
<div class="os-raise-ib-pset-problem" data-content-id="482c9c67-5fb3-42e8-9dfa-474eb2895614" data-problem-comparator="integer" data-problem-type="input" data-solution="4">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<li>How many feet does it fall in 0.5 seconds?</li>
</ol>
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! It falls 4 feet in 0.5 seconds because \(16 (0.5)^2=4\).</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 4. It falls 4 feet in 0.5 seconds because \(16 (0.5)^2=4\).</p>
</div>
</div>
<!--END QUESTION.-->
<!--Do not edit below line.-->
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<ol class="os-raise-noindent" start="2">
<li>To find out where the object is the first few seconds after it was dropped, Elena and Diego created different
tables.</li>
</ol>
<p>Elena’s table:</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">
Time (Seconds)
</th>
<th scope="col">
Distance Fallen (Feet)
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
0
</td>
<td>
0
</td>
</tr>
<tr>
<td>
1
</td>
<td>
16
</td>
</tr>
<tr>
<td>
2
</td>
<td>
64
</td>
</tr>
<tr>
<td>
3
</td>
<td></td>
</tr>
<tr>
<td>
4
</td>
<td></td>
</tr>
<tr>
<td>
\(t\)
</td>
<td></td>
</tr>
</tbody>
</table>
<br>
<p>Diego’s table:</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">
Time (Seconds)
</th>
<th scope="col">
Distance from the Ground (Feet)
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
0
</td>
<td>
576
</td>
</tr>
<tr>
<td>
1
</td>
<td>
560
</td>
</tr>
<tr>
<td>
2
</td>
<td>
512
</td>
</tr>
<tr>
<td>
3
</td>
<td></td>
</tr>
<tr>
<td>
4
</td>
<td></td>
</tr>
<tr>
<td>
t
</td>
<td></td>
</tr>
</tbody>
</table>
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="5283ae4e-4b73-49dc-9a4b-225d4749e3bf" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> How are the two tables alike?</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>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.</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="41b09e9b-75ee-488b-81cd-20d05e338d91" data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> How are the two tables different?</li>
</ol>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>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.</p>
</div>
</div>
<!--Interaction End -->
<br>
<ol class="os-raise-noindent" start="3">
<li>Complete Elena’s table. Be prepared to explain your reasoning.</li>
</ol>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="80db0f18-5142-456c-8c73-6f1987ef7659" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q3a-->
<div class="os-raise-ib-pset-problem" data-content-id="07cfb666-b9a1-4cc5-b6a4-78e440bd664e" data-problem-comparator="integer" data-problem-type="input" data-solution="144">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> What is the distance the object has fallen 3 seconds after it was dropped?</li>
</ol>
</ol>
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 144. The object has fallen 144 feet 3 seconds after it was dropped.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 144. The object has fallen 144 feet 3 seconds after it was dropped.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q3b-->
<div class="os-raise-ib-pset-problem" data-content-id="9cb0e6c1-2f5c-4b7f-b400-addd69c11b52" data-problem-comparator="integer" data-problem-type="input" data-solution="256">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> What is the distance the object has fallen 4 seconds after it was dropped?</li>
</ol>
</ol>
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 256. The object has fallen 256 feet 4 seconds after it was dropped.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 256. The object has fallen 256 feet 4 seconds after it was dropped.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Do not edit below line.-->
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3" type="a">
<li> What is the distance the object has fallen \(t\) seconds after it was dropped? </li>
</ol>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work. </p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1">
<p>Compare your answer:</p>
<p>\(16t^2\)</p>
</div>
<br>
<ol class="os-raise-noindent" start="4">
<li>Complete Diego’s table. Be prepared to explain your reasoning.</li>
</ol>
<div class="os-raise-ib-pset" data-button-text="Check" data-content-id="fa05d1dd-193e-4da8-b658-f8ad4d44021c" data-fire-learning-opportunity-event="eventnameY" data-fire-success-event="eventnameX" data-retry-limit="0" data-schema-version="1.0">
<!--Q4a-->
<div class="os-raise-ib-pset-problem" data-content-id="f680c9cc-d48d-4627-921e-1837b60ec9f1" data-problem-comparator="integer" data-problem-type="input" data-solution="432">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> What is the object’s distance from the ground 3 seconds after it was dropped?</li>
</ol>
</ol>
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 432. The object is 432 feet from the ground 3 seconds after it was dropped.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 432. The object is 432 feet from the ground 3 seconds after it was dropped.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Q4b-->
<div class="os-raise-ib-pset-problem" data-content-id="fea13522-15ea-4820-b214-42f58d143dad" data-problem-comparator="integer" data-problem-type="input" data-solution="320">
<div class="os-raise-ib-pset-problem-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="2" type="a">
<li> What is the object’s distance from the ground 4 seconds after it was dropped?</li>
</ol>
</ol>
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-pset-correct-response">
<p>Correct! 320. The object is 320 feet from the ground 4 seconds after it was dropped.</p>
</div>
<div class="os-raise-ib-pset-attempts-exhausted-response">
<p>The correct answer is 320. The object is 320 feet from the ground 4 seconds after it was dropped.</p>
</div>
</div>
<!--END QUESTION.-->
<!--Do not edit below line.-->
<div class="os-raise-ib-pset-correct-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-pset-encourage-response">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal2" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3" type="a">
<li> What is the object’s distance from the ground \(t\) seconds after it was dropped? </li>
</ol>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work. </p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal2">
<p>Compare your Answer:</p>
<p>\(576-16t^2\)</p>
</div>
<br>
<br>
<h4>Video: Distance as a Quadratic Function of Elapsed Time</h4>
<p>Watch the following video to learn more about quadratic functions.</p>
<div class="os-raise-d-flex-nowrap os-raise-justify-content-center">
<div class="os-raise-video-container"><video controls="true" crossorigin="anonymous">
<source src="https://k12.openstax.org/contents/raise/resources/005a0fd918a3812ba158602b14d462d900fd0661">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/99b550bc7dd378abebde6439ebf81a0f0c4fedd3" srclang="en_us">
https://k12.openstax.org/contents/raise/resources/005a0fd918a3812ba158602b14d462d900fd0661
</video></div>
</div>