Introduction to Aerospace Engineering: Astronautics and Human Spaceflight

<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="aef0b126ed2211efbea212d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+16.00x+2T2019" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+16.00x+2T2019+type@vertical+block@9e6b4bcd9fdc4baea2a330409c88f07f"> <h2 class="hd hd-2 unit-title">The Ideal Rocket Equation</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+16.00x+2T2019+type@html+block@682352deac284a349d1d60bfe5d44e11"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="aef0b126ed2211efbea212d4917dea95" data-runtime-version="1" data-block-type="html" data-course-id="course-v1:MITx+16.00x+2T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+16.00x+2T2019+type@html+block@682352deac284a349d1d60bfe5d44e11"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <h2>2.3.1 The Ideal Rocket Equation</h2> <p><button border-radius="2px" class="mo_button" onclick="window.location.href='/courses/course-v1:MITx+16.00x+3T2016/5e0e82a56b234a2aaf12ac0b3359c3c5#MO23';" style="cursor: pointer;" title="MO 2.3: Apply the Ideal Rocket Equation to calculate the performance of single- and multi-stage rockets" type="button" id="LD">MO 2.3</button></p> <p style="text-align: center;"><em><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;"><img src="/assets/courseware/v1/fd2026c340b5742325c26572bddd12e9/asset-v1:MITx+16.00x+2T2019+type@asset+block/STS114Launch400.png" alt="Launch of Space Shuttle Discovery on STS-114 " type="saveimage" target="[object Object]" height="500" width="400" /></span></em></p> <p style="text-align: center;"><em><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">Launch of Space Shuttle Discovery on STS-114 (Source: NASA)</span></em></p> <p></p> <p>In this section, we will learn why it is so expensive to launch payloads into space.  In particular, we will discuss the various components of mass on a rocket and use these quantities to derive the Rocket Equation in free space (also known as the Ideal Rocket Equation or simply the Rocket Equation).</p> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+16.00x+2T2019+type@video+block@6a5176a1e2af4513aa7140e43cd55e2b"> <div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="aef0b126ed2211efbea212d4917dea95" data-runtime-version="1" data-block-type="video" data-course-id="course-v1:MITx+16.00x+2T2019" data-init="XBlockToXModuleShim" 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="aef0b126ed2211efbea212d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+16.00x+2T2019" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+16.00x+2T2019+type@vertical+block@3eba51ca3dda4f23b4c679ac74576d47"> <h2 class="hd hd-2 unit-title">Implications of the Rocket Equation</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+16.00x+2T2019+type@html+block@95c92469f3764ad9af21a8634b29de19"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="aef0b126ed2211efbea212d4917dea95" data-runtime-version="1" data-block-type="html" data-course-id="course-v1:MITx+16.00x+2T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+16.00x+2T2019+type@html+block@95c92469f3764ad9af21a8634b29de19"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <h2>2.3.2 Implications of the Rocket Equation</h2> <p><button border-radius="2px" class="mo_button" onclick="window.location.href='/courses/course-v1:MITx+16.00x+3T2016/5e0e82a56b234a2aaf12ac0b3359c3c5#MO23';" style="cursor: pointer;" title="MO 2.3: Apply the Ideal Rocket Equation to calculate the performance of single- and multi-stage rockets" type="button" id="LD">MO 2.3</button></p> <p></p> <p style="float: left; margin: 20px;"><img src="/assets/courseware/v1/6ce4b2152c8e204ba04138408b21cbe3/asset-v1:MITx+16.00x+2T2019+type@asset+block/Tsiolkovsky200.gif" alt="Konstantin Tsiolkovsky" type="saveimage" target="[object Object]" /></p> <p>The Rocket Equation was developed by the Russian scientist Konstantin Tsiolkovsky, and uses the principle of conservation of momentum to relate the maximum change in velocity (\(\Delta V\)) to the propellant exhaust velocity and the initial and final rocket mass (\(m_i\) and \(m_f\), respectively). Using just basic physics and math (with a little bit of calculus), the Rocket Equation expresses one of the most fundamental relations in all of astronautics and has significant implications for spaceflight.</p> <p>As you watch the following video, pay special attention to what the various quantities in the Rocket Equation imply for propellant fraction and rocket performance.  You will be asked to apply what you have learned to complete several problems.</p> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+16.00x+2T2019+type@video+block@ae40cc9a53364d349c5c5d9b6425c316"> <div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="aef0b126ed2211efbea212d4917dea95" data-runtime-version="1" data-block-type="video" data-course-id="course-v1:MITx+16.00x+2T2019" data-init="XBlockToXModuleShim" 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="aef0b126ed2211efbea212d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+16.00x+2T2019" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+16.00x+2T2019+type@vertical+block@8ea6140b41a5406486f4142e23e5e4b9"> <h2 class="hd hd-2 unit-title">Review of Concepts: The Ideal Rocket Equation</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+16.00x+2T2019+type@html+block@de3483ef5c7c4f74967db63fe5d64013"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="aef0b126ed2211efbea212d4917dea95" data-runtime-version="1" data-block-type="html" data-course-id="course-v1:MITx+16.00x+2T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+16.00x+2T2019+type@html+block@de3483ef5c7c4f74967db63fe5d64013"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <h2>2.3.3 Review of Concepts: The Ideal Rocket Equation (EQ)</h2> <p><button border-radius="2px" class="mo_button" onclick="window.location.href='/courses/course-v1:MITx+16.00x+3T2016/5e0e82a56b234a2aaf12ac0b3359c3c5#MO23';" style="cursor: pointer;" title="MO 2.3: Apply the Ideal Rocket Equation to calculate the performance of single- and multi-stage rockets" type="button" id="LD">MO 2.3</button></p> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+16.00x+2T2019+type@problem+block@eb9714ad6c6c4499886b66692a4e53ad"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="aef0b126ed2211efbea212d4917dea95" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+16.00x+2T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+16.00x+2T2019+type@problem+block@eb9714ad6c6c4499886b66692a4e53ad"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_eb9714ad6c6c4499886b66692a4e53ad" class="problems-wrapper" role="group" aria-labelledby="eb9714ad6c6c4499886b66692a4e53ad-problem-title" data-problem-id="block-v1:MITx+16.00x+2T2019+type@problem+block@eb9714ad6c6c4499886b66692a4e53ad" data-url="/courses/course-v1:MITx+16.00x+2T2019/xblock/block-v1:MITx+16.00x+2T2019+type@problem+block@eb9714ad6c6c4499886b66692a4e53ad/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="2.0" data-attempts-used="0" data-content=" <h3 class="hd hd-3 problem-header" id="eb9714ad6c6c4499886b66692a4e53ad-problem-title" aria-describedby="block-v1:MITx+16.00x+2T2019+type@problem+block@eb9714ad6c6c4499886b66692a4e53ad-problem-progress" tabindex="-1"> Mass of a Rocket </h3> <div class="problem-progress" id="block-v1:MITx+16.00x+2T2019+type@problem+block@eb9714ad6c6c4499886b66692a4e53ad-problem-progress"></div> <div class="problem"> <div> <p>According to the Ideal Rocket Equation, what components comprise...</p> <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_eb9714ad6c6c4499886b66692a4e53ad_2_1"> <fieldset aria-describedby="status_eb9714ad6c6c4499886b66692a4e53ad_2_1"> <legend id="eb9714ad6c6c4499886b66692a4e53ad_2_1-legend" class="response-fieldset-legend field-group-hd">... the rocket's initial mass \(m_i\)? Check all that apply.</legend> <div class="field"> <input type="checkbox" name="input_eb9714ad6c6c4499886b66692a4e53ad_2_1[]" id="input_eb9714ad6c6c4499886b66692a4e53ad_2_1_choice_0" class="field-input input-checkbox" value="choice_0"/><label id="eb9714ad6c6c4499886b66692a4e53ad_2_1-choice_0-label" for="input_eb9714ad6c6c4499886b66692a4e53ad_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_eb9714ad6c6c4499886b66692a4e53ad_2_1"> Payload </label> </div> <div class="field"> <input type="checkbox" name="input_eb9714ad6c6c4499886b66692a4e53ad_2_1[]" id="input_eb9714ad6c6c4499886b66692a4e53ad_2_1_choice_1" class="field-input input-checkbox" value="choice_1"/><label id="eb9714ad6c6c4499886b66692a4e53ad_2_1-choice_1-label" for="input_eb9714ad6c6c4499886b66692a4e53ad_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_eb9714ad6c6c4499886b66692a4e53ad_2_1"> Propellant </label> </div> <div class="field"> <input type="checkbox" name="input_eb9714ad6c6c4499886b66692a4e53ad_2_1[]" id="input_eb9714ad6c6c4499886b66692a4e53ad_2_1_choice_2" class="field-input input-checkbox" value="choice_2"/><label id="eb9714ad6c6c4499886b66692a4e53ad_2_1-choice_2-label" for="input_eb9714ad6c6c4499886b66692a4e53ad_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_eb9714ad6c6c4499886b66692a4e53ad_2_1"> Structure </label> </div> <span id="answer_eb9714ad6c6c4499886b66692a4e53ad_2_1"/> </fieldset> <div class="indicator-container"> <span class="status unanswered" id="status_eb9714ad6c6c4499886b66692a4e53ad_2_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/> </span> </div> </div></div> <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_eb9714ad6c6c4499886b66692a4e53ad_3_1"> <fieldset aria-describedby="status_eb9714ad6c6c4499886b66692a4e53ad_3_1"> <legend id="eb9714ad6c6c4499886b66692a4e53ad_3_1-legend" class="response-fieldset-legend field-group-hd">... the rocket's final mass \(m_f\)? 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Recall that the relationship between specific impulse and exhaust velocity is \( V_\text{ex} = g_0\, I_\text{sp} \). </p> <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="choicegroup capa_inputtype" id="inputtype_6160014513e94485a136db9f2e1239df_2_1"> <fieldset aria-describedby="status_6160014513e94485a136db9f2e1239df_2_1"> <legend id="6160014513e94485a136db9f2e1239df_2_1-legend" class="response-fieldset-legend field-group-hd">If one engine fails, what is the impact on the total \(\Delta V\) that can be obtained from the system?</legend> <div class="field"> <input type="radio" name="input_6160014513e94485a136db9f2e1239df_2_1" id="input_6160014513e94485a136db9f2e1239df_2_1_choice_0" class="field-input input-radio" value="choice_0"/><label id="6160014513e94485a136db9f2e1239df_2_1-choice_0-label" for="input_6160014513e94485a136db9f2e1239df_2_1_choice_0" class="response-label field-label label-inline" aria-describedby="status_6160014513e94485a136db9f2e1239df_2_1"> \(\Delta V\) will double </label> </div> <div class="field"> <input type="radio" name="input_6160014513e94485a136db9f2e1239df_2_1" id="input_6160014513e94485a136db9f2e1239df_2_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="6160014513e94485a136db9f2e1239df_2_1-choice_1-label" for="input_6160014513e94485a136db9f2e1239df_2_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_6160014513e94485a136db9f2e1239df_2_1"> \(\Delta V\) will be cut in half </label> </div> <div class="field"> <input type="radio" name="input_6160014513e94485a136db9f2e1239df_2_1" id="input_6160014513e94485a136db9f2e1239df_2_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="6160014513e94485a136db9f2e1239df_2_1-choice_2-label" for="input_6160014513e94485a136db9f2e1239df_2_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_6160014513e94485a136db9f2e1239df_2_1"> \(\Delta V\) will stay the same </label> </div> <span id="answer_6160014513e94485a136db9f2e1239df_2_1"/> </fieldset> <div class="indicator-container"> <span class="status unanswered" id="status_6160014513e94485a136db9f2e1239df_2_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/> </span> </div> </div></div> <div class="solution-span"> <span id="solution_6160014513e94485a136db9f2e1239df_solution_1"/> </div></div> <div class="action"> <input type="hidden" name="problem_id" value="Rocket Propulsion Conceptual Question" /> <div class="submit-attempt-container"> <button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_6160014513e94485a136db9f2e1239df" > <span class="submit-label">Submit</span> </button> <div class="submission-feedback" id="submission_feedback_6160014513e94485a136db9f2e1239df"> <span class="sr">Some problems have options such as save, reset, hints, or show answer. 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="aef0b126ed2211efbea212d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+16.00x+2T2019" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+16.00x+2T2019+type@vertical+block@da7dc61b213a41b7bfc4c317440fdbb2"> <h2 class="hd hd-2 unit-title">Rocket Staging</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+16.00x+2T2019+type@html+block@347548878b804ecea147e1f5e611da83"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="aef0b126ed2211efbea212d4917dea95" data-runtime-version="1" data-block-type="html" data-course-id="course-v1:MITx+16.00x+2T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+16.00x+2T2019+type@html+block@347548878b804ecea147e1f5e611da83"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <h2>2.3.4 Rocket Staging</h2> <p><button border-radius="2px" class="mo_button" onclick="window.location.href='/courses/course-v1:MITx+16.00x+3T2016/5e0e82a56b234a2aaf12ac0b3359c3c5#MO23';" style="cursor: pointer;" title="MO 2.3: Apply the Ideal Rocket Equation to calculate the performance of single- and multi-stage rockets" type="button" id="LD">MO 2.3</button> <button border-radius="2px" class="mo_button" onclick="window.location.href='/courses/course-v1:MITx+16.00x+3T2016/5e0e82a56b234a2aaf12ac0b3359c3c5#MO24';" style="cursor: pointer;" title="MO 2.4: Explain why rocket staging is important for achieving Earth orbit" type="button" id="LD">MO 2.4</button></p> <p><img style="display: block; margin: 20px auto 20px auto;" src="/assets/courseware/v1/f00f09c8bfeeb437b647b9442d7078e8/asset-v1:MITx+16.00x+2T2019+type@asset+block/Apollo11Staging300.png" alt="1st stage separation of Saturn V on Apollo 11 mission" type="saveimage" target="[object Object]" /></p> <p></p> <p style="text-align: center;"><em><span style="font-family: 'Open Sans', Verdana, Arial, Helvetica, sans-serif;">First-stage separation of Saturn V rocket during the Apollo 11 mission (Image credit: NASA)</span></em></p> <p></p> <p>Now that we are familiar with the Ideal Rocket Equation, we will discuss another important aspect of rocket propulsion: rocket staging. As we will see, achieving Earth orbit requires a very large \(\Delta V\) and a considerable propellant mass fraction --- so much so that it is unrealistic to do this with a single-stage rocket. In a multi-stage rocket, once the propellant in the first stage is consumed, the first stage structure is discarded, and the rocket continues onward using propellant from the second stage; once the second stage propellant is exhausted, the second stage structure is discarded, and the rocket continues onward using the third stage propellant, and so on. This approach allows the rocket to achieve a larger \(\Delta V\) for the same initial mass at launch.</p>  <p>After watching following video about rocket staging, you will have an opportunity to apply what you learned to solve a two-stage rocket problem.</p> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+16.00x+2T2019+type@video+block@4cde424d03e24de6874d67ab6abd7eff"> <div class="xblock xblock-public_view xblock-public_view-video xmodule_display xmodule_VideoBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="aef0b126ed2211efbea212d4917dea95" data-runtime-version="1" data-block-type="video" data-course-id="course-v1:MITx+16.00x+2T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+16.00x+2T2019+type@video+block@4cde424d03e24de6874d67ab6abd7eff"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Video"} </script> <h3 class="hd hd-2">Video: Rocket Staging</h3> <div 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<div class="xblock xblock-public_view xblock-public_view-vertical" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="aef0b126ed2211efbea212d4917dea95" data-runtime-version="1" data-block-type="vertical" data-course-id="course-v1:MITx+16.00x+2T2019" data-init="VerticalStudentView" data-usage-id="block-v1:MITx+16.00x+2T2019+type@vertical+block@7d1333d52ce34f36ba162290b81b28f1"> <h2 class="hd hd-2 unit-title">Review of Concepts: Rocket Staging</h2> <div class="vert-mod"> <div class="vert vert-0" data-id="block-v1:MITx+16.00x+2T2019+type@html+block@ae4861b1699149b5b1e53edf5eb72599"> <div class="xblock xblock-public_view xblock-public_view-html xmodule_display xmodule_HtmlBlock" data-has-score="False" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="aef0b126ed2211efbea212d4917dea95" data-runtime-version="1" data-block-type="html" data-course-id="course-v1:MITx+16.00x+2T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+16.00x+2T2019+type@html+block@ae4861b1699149b5b1e53edf5eb72599"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "HTMLModule"} </script> <h2>2.3.5 Review of Concepts: Rocket Staging</h2> <p><button border-radius="2px" class="mo_button" onclick="window.location.href='/courses/course-v1:MITx+16.00x+3T2016/5e0e82a56b234a2aaf12ac0b3359c3c5#MO23';" style="cursor: pointer;" title="MO 2.3: Apply the Ideal Rocket Equation to calculate the performance of single- and multi-stage rockets" type="button" id="LD">MO 2.3</button> <button border-radius="2px" class="mo_button" onclick="window.location.href='/courses/course-v1:MITx+16.00x+3T2016/5e0e82a56b234a2aaf12ac0b3359c3c5#MO24';" style="cursor: pointer;" title="MO 2.4: Explain why rocket staging is important for achieving Earth orbit" type="button" id="LD">MO 2.4</button></p> </div> </div> <div class="vert vert-1" data-id="block-v1:MITx+16.00x+2T2019+type@problem+block@783476bfd93e455b8b58119a203c7642"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="aef0b126ed2211efbea212d4917dea95" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+16.00x+2T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+16.00x+2T2019+type@problem+block@783476bfd93e455b8b58119a203c7642"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_783476bfd93e455b8b58119a203c7642" class="problems-wrapper" role="group" aria-labelledby="783476bfd93e455b8b58119a203c7642-problem-title" data-problem-id="block-v1:MITx+16.00x+2T2019+type@problem+block@783476bfd93e455b8b58119a203c7642" data-url="/courses/course-v1:MITx+16.00x+2T2019/xblock/block-v1:MITx+16.00x+2T2019+type@problem+block@783476bfd93e455b8b58119a203c7642/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="3.0" data-attempts-used="0" data-content=" <h3 class="hd hd-3 problem-header" id="783476bfd93e455b8b58119a203c7642-problem-title" aria-describedby="block-v1:MITx+16.00x+2T2019+type@problem+block@783476bfd93e455b8b58119a203c7642-problem-progress" tabindex="-1"> Part 1: Single Stage </h3> <div class="problem-progress" id="block-v1:MITx+16.00x+2T2019+type@problem+block@783476bfd93e455b8b58119a203c7642-problem-progress"></div> <div class="problem"> <div> <p>Consider a rocket carrying 100,000 kg of propellant, 10,000 kg of structure, and 5000 kg of payload.</p> <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_783476bfd93e455b8b58119a203c7642_2_1" class="inputtype formulaequationinput"> <div class="unanswered"> <label class="problem-group-label" for="input_783476bfd93e455b8b58119a203c7642_2_1" id="label_783476bfd93e455b8b58119a203c7642_2_1">What is the propellant fraction of this rocket? Round your answer to at least 2 decimal places (i.e., enter 0.25 to represent 25%)</label> <input type="text" name="input_783476bfd93e455b8b58119a203c7642_2_1" id="input_783476bfd93e455b8b58119a203c7642_2_1" data-input-id="783476bfd93e455b8b58119a203c7642_2_1" value="" aria-describedby="status_783476bfd93e455b8b58119a203c7642_2_1" size="20"/> <span class="trailing_text" id="trailing_text_783476bfd93e455b8b58119a203c7642_2_1"/> <span class="status unanswered" id="status_783476bfd93e455b8b58119a203c7642_2_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/> </span> <p id="answer_783476bfd93e455b8b58119a203c7642_2_1" class="answer"/> <div id="input_783476bfd93e455b8b58119a203c7642_2_1_preview" class="equation"> \(\) <img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/> </div> </div> <div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/> </div></div> <div class="solution-span"> <span id="solution_783476bfd93e455b8b58119a203c7642_solution_1"/> </div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_783476bfd93e455b8b58119a203c7642_3_1" class="inputtype formulaequationinput"> <div class="unanswered"> <label class="problem-group-label" for="input_783476bfd93e455b8b58119a203c7642_3_1" id="label_783476bfd93e455b8b58119a203c7642_3_1">What is the payload fraction of this rocket? Round your answer to at least 2 decimal places.</label> <input type="text" name="input_783476bfd93e455b8b58119a203c7642_3_1" id="input_783476bfd93e455b8b58119a203c7642_3_1" data-input-id="783476bfd93e455b8b58119a203c7642_3_1" value="" aria-describedby="status_783476bfd93e455b8b58119a203c7642_3_1" size="20"/> <span class="trailing_text" id="trailing_text_783476bfd93e455b8b58119a203c7642_3_1"/> <span class="status unanswered" id="status_783476bfd93e455b8b58119a203c7642_3_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/> </span> <p id="answer_783476bfd93e455b8b58119a203c7642_3_1" class="answer"/> <div id="input_783476bfd93e455b8b58119a203c7642_3_1_preview" class="equation"> \(\) <img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/> </div> </div> <div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/> </div></div> <div class="solution-span"> <span id="solution_783476bfd93e455b8b58119a203c7642_solution_2"/> </div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_783476bfd93e455b8b58119a203c7642_4_1" class="inputtype formulaequationinput"> <div class="unanswered"> <label class="problem-group-label" for="input_783476bfd93e455b8b58119a203c7642_4_1" id="label_783476bfd93e455b8b58119a203c7642_4_1">Recall that the relationship between specific impulse and exhaust velocity is: \( V_\text{ex} = g_0\, I_\text{sp} \). Assuming that \(I_\text{sp} = 450\,\text{s}\) and \(g_{0} = 9.81\,\text{m/s}^2\), what is the value of \(\Delta V\) in m/s that this rocket will produce if all the propellant is consumed in one stage?</label> <input type="text" name="input_783476bfd93e455b8b58119a203c7642_4_1" id="input_783476bfd93e455b8b58119a203c7642_4_1" data-input-id="783476bfd93e455b8b58119a203c7642_4_1" value="" aria-describedby="status_783476bfd93e455b8b58119a203c7642_4_1" size="20"/> <span class="trailing_text" id="trailing_text_783476bfd93e455b8b58119a203c7642_4_1"/> <span class="status unanswered" id="status_783476bfd93e455b8b58119a203c7642_4_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/> </span> <p id="answer_783476bfd93e455b8b58119a203c7642_4_1" class="answer"/> <div id="input_783476bfd93e455b8b58119a203c7642_4_1_preview" class="equation"> \(\) <img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/> </div> </div> <div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/> </div></div> <div class="solution-span"> <span id="solution_783476bfd93e455b8b58119a203c7642_solution_3"/> </div></div> <div class="action"> <input type="hidden" name="problem_id" value="Part 1: Single Stage" /> <div class="submit-attempt-container"> <button type="button" class="submit btn-brand" data-submitting="Submitting" data-value="Submit" data-should-enable-submit-button="True" aria-describedby="submission_feedback_783476bfd93e455b8b58119a203c7642" > <span class="submit-label">Submit</span> </button> <div class="submission-feedback" id="submission_feedback_783476bfd93e455b8b58119a203c7642"> <span class="sr">Some problems have options such as save, reset, hints, or show answer. These options follow the Submit button.</span> </div> </div> <div class="problem-action-buttons-wrapper"> </div> </div> <div class="notification warning notification-gentle-alert is-hidden" tabindex="-1"> <span class="icon fa fa-exclamation-circle" aria-hidden="true"></span> <span class="notification-message" aria-describedby="783476bfd93e455b8b58119a203c7642-problem-title"> </span> <div class="notification-btn-wrapper"> <button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button> </div> </div> <div class="notification warning notification-save is-hidden" tabindex="-1"> <span class="icon fa fa-save" aria-hidden="true"></span> <span class="notification-message" aria-describedby="783476bfd93e455b8b58119a203c7642-problem-title">None </span> <div class="notification-btn-wrapper"> <button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button> </div> </div> <div class="notification general notification-show-answer is-hidden" tabindex="-1"> <span class="icon fa fa-info-circle" aria-hidden="true"></span> <span class="notification-message" aria-describedby="783476bfd93e455b8b58119a203c7642-problem-title">Answers are displayed within the problem </span> <div class="notification-btn-wrapper"> <button type="button" class="btn btn-default btn-small notification-btn review-btn sr">Review</button> </div> </div> </div> " data-graded="True"> <p class="loading-spinner"> <i class="fa fa-spinner fa-pulse fa-2x fa-fw"></i> <span class="sr">Loading…</span> </p> </div> </div> </div> <div class="vert vert-2" data-id="block-v1:MITx+16.00x+2T2019+type@problem+block@05280c185a36497d92f6d4d1405fb290"> <div class="xblock xblock-public_view xblock-public_view-problem xmodule_display xmodule_ProblemBlock" data-has-score="True" data-runtime-class="LmsRuntime" data-graded="True" data-request-token="aef0b126ed2211efbea212d4917dea95" data-runtime-version="1" data-block-type="problem" data-course-id="course-v1:MITx+16.00x+2T2019" data-init="XBlockToXModuleShim" data-usage-id="block-v1:MITx+16.00x+2T2019+type@problem+block@05280c185a36497d92f6d4d1405fb290"> <script type="json/xblock-args" class="xblock-json-init-args"> {"xmodule-type": "Problem"} </script> <div id="problem_05280c185a36497d92f6d4d1405fb290" class="problems-wrapper" role="group" aria-labelledby="05280c185a36497d92f6d4d1405fb290-problem-title" data-problem-id="block-v1:MITx+16.00x+2T2019+type@problem+block@05280c185a36497d92f6d4d1405fb290" data-url="/courses/course-v1:MITx+16.00x+2T2019/xblock/block-v1:MITx+16.00x+2T2019+type@problem+block@05280c185a36497d92f6d4d1405fb290/handler/xmodule_handler" data-problem-score="0.0" data-problem-total-possible="4.0" data-attempts-used="0" data-content=" <h3 class="hd hd-3 problem-header" id="05280c185a36497d92f6d4d1405fb290-problem-title" aria-describedby="block-v1:MITx+16.00x+2T2019+type@problem+block@05280c185a36497d92f6d4d1405fb290-problem-progress" tabindex="-1"> Part 2: Multi-Stage </h3> <div class="problem-progress" id="block-v1:MITx+16.00x+2T2019+type@problem+block@05280c185a36497d92f6d4d1405fb290-problem-progress"></div> <div class="problem"> <div> <p>Now break the rocket into two equal stages, with each stage containing half the propellant and half the structure of Part 1 (i.e., 50,000 kg of propellant and 5000 kg of structure in each stage). The 5000 kg of payload goes on the front of the second stage.</p> <p>Note: A diagram might be useful to help you keep track of the various quantities. Make sure you are very clear in your work about what variables correspond to what quantities. Remember that \(m_f\) of the first stage includes the first stage structure, but \(m_i\) of the second stage does not include the first stage structure, since that is lost during staging.</p> <div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div id="formulaequationinput_05280c185a36497d92f6d4d1405fb290_2_1" class="inputtype formulaequationinput"> <div class="unanswered"> <label class="problem-group-label" for="input_05280c185a36497d92f6d4d1405fb290_2_1" id="label_05280c185a36497d92f6d4d1405fb290_2_1">Calculate \(\Delta V\) from the first stage in m/s:</label> <input type="text" name="input_05280c185a36497d92f6d4d1405fb290_2_1" id="input_05280c185a36497d92f6d4d1405fb290_2_1" data-input-id="05280c185a36497d92f6d4d1405fb290_2_1" value="" aria-describedby="status_05280c185a36497d92f6d4d1405fb290_2_1" size="20"/> <span class="trailing_text" id="trailing_text_05280c185a36497d92f6d4d1405fb290_2_1"/> <span class="status unanswered" id="status_05280c185a36497d92f6d4d1405fb290_2_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/> </span> <p id="answer_05280c185a36497d92f6d4d1405fb290_2_1" class="answer"/> <div id="input_05280c185a36497d92f6d4d1405fb290_2_1_preview" class="equation"> \(\) <img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/> </div> </div> <div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/> </div></div> <div class="solution-span"> <span id="solution_05280c185a36497d92f6d4d1405fb290_solution_1"/> </div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div id="formulaequationinput_05280c185a36497d92f6d4d1405fb290_3_1" class="inputtype formulaequationinput"> <div class="unanswered"> <label class="problem-group-label" for="input_05280c185a36497d92f6d4d1405fb290_3_1" id="label_05280c185a36497d92f6d4d1405fb290_3_1">Calculate \(\Delta V\) from the second stage in m/s:</label> <input type="text" name="input_05280c185a36497d92f6d4d1405fb290_3_1" id="input_05280c185a36497d92f6d4d1405fb290_3_1" data-input-id="05280c185a36497d92f6d4d1405fb290_3_1" value="" aria-describedby="status_05280c185a36497d92f6d4d1405fb290_3_1" size="20"/> <span class="trailing_text" id="trailing_text_05280c185a36497d92f6d4d1405fb290_3_1"/> <span class="status unanswered" id="status_05280c185a36497d92f6d4d1405fb290_3_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/> </span> <p id="answer_05280c185a36497d92f6d4d1405fb290_3_1" class="answer"/> <div id="input_05280c185a36497d92f6d4d1405fb290_3_1_preview" class="equation"> \(\) <img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/> </div> </div> <div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/> </div></div> <div class="solution-span"> <span id="solution_05280c185a36497d92f6d4d1405fb290_solution_2"/> </div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 3" role="group"><div id="formulaequationinput_05280c185a36497d92f6d4d1405fb290_4_1" class="inputtype formulaequationinput"> <div class="unanswered"> <label class="problem-group-label" for="input_05280c185a36497d92f6d4d1405fb290_4_1" id="label_05280c185a36497d92f6d4d1405fb290_4_1">Add the two values of \(\Delta V\) together to get \(\Delta V_\text{total}\) in m/s for the two-stage rocket:</label> <input type="text" name="input_05280c185a36497d92f6d4d1405fb290_4_1" id="input_05280c185a36497d92f6d4d1405fb290_4_1" data-input-id="05280c185a36497d92f6d4d1405fb290_4_1" value="" aria-describedby="status_05280c185a36497d92f6d4d1405fb290_4_1" size="20"/> <span class="trailing_text" id="trailing_text_05280c185a36497d92f6d4d1405fb290_4_1"/> <span class="status unanswered" id="status_05280c185a36497d92f6d4d1405fb290_4_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/> </span> <p id="answer_05280c185a36497d92f6d4d1405fb290_4_1" class="answer"/> <div id="input_05280c185a36497d92f6d4d1405fb290_4_1_preview" class="equation"> \(\) <img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/> </div> </div> <div class="script_placeholder" data-src="/static/js/capa/src/formula_equation_preview.b1967ab28c31.js"/> </div></div> <div class="solution-span"> <span id="solution_05280c185a36497d92f6d4d1405fb290_solution_3"/> </div><div class="wrapper-problem-response" tabindex="-1" aria-label="Question 4" role="group"><div id="formulaequationinput_05280c185a36497d92f6d4d1405fb290_5_1" class="inputtype formulaequationinput"> <div class="unanswered"> <label class="problem-group-label" for="input_05280c185a36497d92f6d4d1405fb290_5_1" id="label_05280c185a36497d92f6d4d1405fb290_5_1">How much extra \(\Delta V\) in m/s is obtained by the two-stage rocket as compared with the single-stage rocket?</label> <input type="text" name="input_05280c185a36497d92f6d4d1405fb290_5_1" id="input_05280c185a36497d92f6d4d1405fb290_5_1" data-input-id="05280c185a36497d92f6d4d1405fb290_5_1" value="" aria-describedby="status_05280c185a36497d92f6d4d1405fb290_5_1" size="20"/> <span class="trailing_text" id="trailing_text_05280c185a36497d92f6d4d1405fb290_5_1"/> <span class="status unanswered" id="status_05280c185a36497d92f6d4d1405fb290_5_1" data-tooltip="Not yet answered."> <span class="sr">unanswered</span><span class="status-icon" aria-hidden="true"/> </span> <p id="answer_05280c185a36497d92f6d4d1405fb290_5_1" class="answer"/> <div id="input_05280c185a36497d92f6d4d1405fb290_5_1_preview" class="equation"> \(\) <img src="/static/images/spinner.bc34f953403f.gif" class="loading" alt="Loading"/> </div> </div> <div class="script_placeholder" 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