A question about free fall, velocity, and the height of an object.












4












$begingroup$


A falling stone is at a certain instant $100$ feet above the ground. Two seconds later it is only $16$ feet above the ground.



a) If it was thrown downward with an initial speed of $5$ ft/sec, from what height was it thrown?



b) If it was thrown upward with an initial speed of $10$ ft/sec, from what height was it thrown?



I got the wrong answers when working on this.



To solve a):



$$s(t+2) - s(t) = 84$$
$$s(t) = v_0t+cfrac{1}{2}at^2, v_0 = 5, a = 32$$
$$left[5(t+2)+16(t+2)^2right]-(5t+16t^2)=84$$
$$64t=10$$
$$t=cfrac{5}{8}$$
$$5left(cfrac{5}{8}right)+16left(cfrac{5}{8}right)^2=9.375$$
$$h_0=109.375$$



To solve b):



$$100=-16t^2+7t+h_0$$
$$16=-16(t+2)^2+7(t+2)+h_0$$
now subtract the smaller constant from the larger
$$-84=-71t+7t-50$$
$$t=cfrac{34}{71}$$
$$100=-16left(cfrac{34}{71}right)^2+7left(cfrac{34}{71}right)+h_0$$
$$h_0=cfrac{505698}{5041}$$



However the answers are:
$a=cfrac{6475}{65}$
$b=100$



What am I doing wrong?










share|cite|improve this question









$endgroup$

















    4












    $begingroup$


    A falling stone is at a certain instant $100$ feet above the ground. Two seconds later it is only $16$ feet above the ground.



    a) If it was thrown downward with an initial speed of $5$ ft/sec, from what height was it thrown?



    b) If it was thrown upward with an initial speed of $10$ ft/sec, from what height was it thrown?



    I got the wrong answers when working on this.



    To solve a):



    $$s(t+2) - s(t) = 84$$
    $$s(t) = v_0t+cfrac{1}{2}at^2, v_0 = 5, a = 32$$
    $$left[5(t+2)+16(t+2)^2right]-(5t+16t^2)=84$$
    $$64t=10$$
    $$t=cfrac{5}{8}$$
    $$5left(cfrac{5}{8}right)+16left(cfrac{5}{8}right)^2=9.375$$
    $$h_0=109.375$$



    To solve b):



    $$100=-16t^2+7t+h_0$$
    $$16=-16(t+2)^2+7(t+2)+h_0$$
    now subtract the smaller constant from the larger
    $$-84=-71t+7t-50$$
    $$t=cfrac{34}{71}$$
    $$100=-16left(cfrac{34}{71}right)^2+7left(cfrac{34}{71}right)+h_0$$
    $$h_0=cfrac{505698}{5041}$$



    However the answers are:
    $a=cfrac{6475}{65}$
    $b=100$



    What am I doing wrong?










    share|cite|improve this question









    $endgroup$















      4












      4








      4





      $begingroup$


      A falling stone is at a certain instant $100$ feet above the ground. Two seconds later it is only $16$ feet above the ground.



      a) If it was thrown downward with an initial speed of $5$ ft/sec, from what height was it thrown?



      b) If it was thrown upward with an initial speed of $10$ ft/sec, from what height was it thrown?



      I got the wrong answers when working on this.



      To solve a):



      $$s(t+2) - s(t) = 84$$
      $$s(t) = v_0t+cfrac{1}{2}at^2, v_0 = 5, a = 32$$
      $$left[5(t+2)+16(t+2)^2right]-(5t+16t^2)=84$$
      $$64t=10$$
      $$t=cfrac{5}{8}$$
      $$5left(cfrac{5}{8}right)+16left(cfrac{5}{8}right)^2=9.375$$
      $$h_0=109.375$$



      To solve b):



      $$100=-16t^2+7t+h_0$$
      $$16=-16(t+2)^2+7(t+2)+h_0$$
      now subtract the smaller constant from the larger
      $$-84=-71t+7t-50$$
      $$t=cfrac{34}{71}$$
      $$100=-16left(cfrac{34}{71}right)^2+7left(cfrac{34}{71}right)+h_0$$
      $$h_0=cfrac{505698}{5041}$$



      However the answers are:
      $a=cfrac{6475}{65}$
      $b=100$



      What am I doing wrong?










      share|cite|improve this question









      $endgroup$




      A falling stone is at a certain instant $100$ feet above the ground. Two seconds later it is only $16$ feet above the ground.



      a) If it was thrown downward with an initial speed of $5$ ft/sec, from what height was it thrown?



      b) If it was thrown upward with an initial speed of $10$ ft/sec, from what height was it thrown?



      I got the wrong answers when working on this.



      To solve a):



      $$s(t+2) - s(t) = 84$$
      $$s(t) = v_0t+cfrac{1}{2}at^2, v_0 = 5, a = 32$$
      $$left[5(t+2)+16(t+2)^2right]-(5t+16t^2)=84$$
      $$64t=10$$
      $$t=cfrac{5}{8}$$
      $$5left(cfrac{5}{8}right)+16left(cfrac{5}{8}right)^2=9.375$$
      $$h_0=109.375$$



      To solve b):



      $$100=-16t^2+7t+h_0$$
      $$16=-16(t+2)^2+7(t+2)+h_0$$
      now subtract the smaller constant from the larger
      $$-84=-71t+7t-50$$
      $$t=cfrac{34}{71}$$
      $$100=-16left(cfrac{34}{71}right)^2+7left(cfrac{34}{71}right)+h_0$$
      $$h_0=cfrac{505698}{5041}$$



      However the answers are:
      $a=cfrac{6475}{65}$
      $b=100$



      What am I doing wrong?







      calculus






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 4 hours ago









      JinzuJinzu

      403513




      403513






















          2 Answers
          2






          active

          oldest

          votes


















          1












          $begingroup$

          The error in a) is simple:



          From $64t=10$ it follows $t=frac5{32} neq frac58$. Substituting this into your formula for $s(t)$ (including that after time $t$ you are at $100$ft) yields:



          $h_0=100+5left(frac58right) + 16left(frac58right)^2=frac{6475}{64}$



          which is very similar to your answer key (I assume you mistyped the denominator).



          In b) you seem to be calculating with $v_0=7ft/s$, but $v_0=10ft/s$ was given.






          share|cite|improve this answer









          $endgroup$





















            1












            $begingroup$

            the solution of
            $$left[5(t+2)+16(t+2)^2right]-(5t+16t^2)=84$$
            should be $t=frac{5}{32}$ not $t=frac{5}{8}$






            share|cite|improve this answer









            $endgroup$














              Your Answer





              StackExchange.ifUsing("editor", function () {
              return StackExchange.using("mathjaxEditing", function () {
              StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
              StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
              });
              });
              }, "mathjax-editing");

              StackExchange.ready(function() {
              var channelOptions = {
              tags: "".split(" "),
              id: "69"
              };
              initTagRenderer("".split(" "), "".split(" "), channelOptions);

              StackExchange.using("externalEditor", function() {
              // Have to fire editor after snippets, if snippets enabled
              if (StackExchange.settings.snippets.snippetsEnabled) {
              StackExchange.using("snippets", function() {
              createEditor();
              });
              }
              else {
              createEditor();
              }
              });

              function createEditor() {
              StackExchange.prepareEditor({
              heartbeatType: 'answer',
              autoActivateHeartbeat: false,
              convertImagesToLinks: true,
              noModals: true,
              showLowRepImageUploadWarning: true,
              reputationToPostImages: 10,
              bindNavPrevention: true,
              postfix: "",
              imageUploader: {
              brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
              contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
              allowUrls: true
              },
              noCode: true, onDemand: true,
              discardSelector: ".discard-answer"
              ,immediatelyShowMarkdownHelp:true
              });


              }
              });














              draft saved

              draft discarded


















              StackExchange.ready(
              function () {
              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3169890%2fa-question-about-free-fall-velocity-and-the-height-of-an-object%23new-answer', 'question_page');
              }
              );

              Post as a guest















              Required, but never shown

























              2 Answers
              2






              active

              oldest

              votes








              2 Answers
              2






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes









              1












              $begingroup$

              The error in a) is simple:



              From $64t=10$ it follows $t=frac5{32} neq frac58$. Substituting this into your formula for $s(t)$ (including that after time $t$ you are at $100$ft) yields:



              $h_0=100+5left(frac58right) + 16left(frac58right)^2=frac{6475}{64}$



              which is very similar to your answer key (I assume you mistyped the denominator).



              In b) you seem to be calculating with $v_0=7ft/s$, but $v_0=10ft/s$ was given.






              share|cite|improve this answer









              $endgroup$


















                1












                $begingroup$

                The error in a) is simple:



                From $64t=10$ it follows $t=frac5{32} neq frac58$. Substituting this into your formula for $s(t)$ (including that after time $t$ you are at $100$ft) yields:



                $h_0=100+5left(frac58right) + 16left(frac58right)^2=frac{6475}{64}$



                which is very similar to your answer key (I assume you mistyped the denominator).



                In b) you seem to be calculating with $v_0=7ft/s$, but $v_0=10ft/s$ was given.






                share|cite|improve this answer









                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  The error in a) is simple:



                  From $64t=10$ it follows $t=frac5{32} neq frac58$. Substituting this into your formula for $s(t)$ (including that after time $t$ you are at $100$ft) yields:



                  $h_0=100+5left(frac58right) + 16left(frac58right)^2=frac{6475}{64}$



                  which is very similar to your answer key (I assume you mistyped the denominator).



                  In b) you seem to be calculating with $v_0=7ft/s$, but $v_0=10ft/s$ was given.






                  share|cite|improve this answer









                  $endgroup$



                  The error in a) is simple:



                  From $64t=10$ it follows $t=frac5{32} neq frac58$. Substituting this into your formula for $s(t)$ (including that after time $t$ you are at $100$ft) yields:



                  $h_0=100+5left(frac58right) + 16left(frac58right)^2=frac{6475}{64}$



                  which is very similar to your answer key (I assume you mistyped the denominator).



                  In b) you seem to be calculating with $v_0=7ft/s$, but $v_0=10ft/s$ was given.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 4 hours ago









                  IngixIngix

                  5,097159




                  5,097159























                      1












                      $begingroup$

                      the solution of
                      $$left[5(t+2)+16(t+2)^2right]-(5t+16t^2)=84$$
                      should be $t=frac{5}{32}$ not $t=frac{5}{8}$






                      share|cite|improve this answer









                      $endgroup$


















                        1












                        $begingroup$

                        the solution of
                        $$left[5(t+2)+16(t+2)^2right]-(5t+16t^2)=84$$
                        should be $t=frac{5}{32}$ not $t=frac{5}{8}$






                        share|cite|improve this answer









                        $endgroup$
















                          1












                          1








                          1





                          $begingroup$

                          the solution of
                          $$left[5(t+2)+16(t+2)^2right]-(5t+16t^2)=84$$
                          should be $t=frac{5}{32}$ not $t=frac{5}{8}$






                          share|cite|improve this answer









                          $endgroup$



                          the solution of
                          $$left[5(t+2)+16(t+2)^2right]-(5t+16t^2)=84$$
                          should be $t=frac{5}{32}$ not $t=frac{5}{8}$







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered 4 hours ago









                          E.H.EE.H.E

                          16.1k11968




                          16.1k11968






























                              draft saved

                              draft discarded




















































                              Thanks for contributing an answer to Mathematics Stack Exchange!


                              • Please be sure to answer the question. Provide details and share your research!

                              But avoid



                              • Asking for help, clarification, or responding to other answers.

                              • Making statements based on opinion; back them up with references or personal experience.


                              Use MathJax to format equations. MathJax reference.


                              To learn more, see our tips on writing great answers.




                              draft saved


                              draft discarded














                              StackExchange.ready(
                              function () {
                              StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3169890%2fa-question-about-free-fall-velocity-and-the-height-of-an-object%23new-answer', 'question_page');
                              }
                              );

                              Post as a guest















                              Required, but never shown





















































                              Required, but never shown














                              Required, but never shown












                              Required, but never shown







                              Required, but never shown

































                              Required, but never shown














                              Required, but never shown












                              Required, but never shown







                              Required, but never shown







                              Popular posts from this blog

                              Liste der Baudenkmale in Friedland (Mecklenburg)

                              Single-Malt-Whisky

                              Czorneboh