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source: "Derivation of Gravitational Potential Energy" by Rhett Allain

scenario description Suppose an
tangible entity object
condition starts
an
location infinite distance
from a
tangible entity moon
and is
action dropped
,
action falling
direction towards
the
tangible entity moon
cause due to
measure gravitational acceleration.
What is the
measure speed
of the
tangible entity object
condition when
tangible entity it
is
measure distance
\(r\) from the
tangible entity moon?
question What is the
measure speed
of the
tangible entity object
condition when
tangible entity it
is
measure distance
\(r\) from the
tangible entity moon?
figure of scenario
Figure 1: small mass falling towards a moon from initial position at infinity.

initial condition The
condition initial condition
is
expression as Content MathML \begin{equation} v(x=\infty) =0 \label{eq:initial_velocity} \end{equation} 
    <?xml version="1.0" encoding="UTF-8"?>
           <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v=0" display="block">
             <apply>
               <eq/>
               <ci>v</ci>
               <cn type="integer">0</cn>
             </apply>
           </math>
The
measure force
acting on the
tangible entity object
is
expression as Content MathML \begin{equation} \vec{F} = \frac{-G m_1 m_2}{x^2} \hat{x} \label{eq:gravitational force} \end{equation} 
    <?xml version="1.0" encoding="UTF-8"?>
     <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\vec{F}=\frac{-Gm_{1}m_{2}}{x^{2}}\hat{x}" display="block">
       <apply>
         <eq/>
         <apply>
           <ci>→</ci>
           <ci>𝐹</ci>
         </apply>
         <apply>
           <times/>
           <apply>
             <divide/>
             <apply>
               <minus/>
               <apply>
                 <times/>
                 <ci>𝐺</ci>
                 <apply>
                   <csymbol cd="ambiguous">subscript</csymbol>
                   <ci>𝑚</ci>
                   <cn type="integer">1</cn>
                 </apply>
                 <apply>
                   <csymbol cd="ambiguous">subscript</csymbol>
                   <ci>𝑚</ci>
                   <cn type="integer">2</cn>
                 </apply>
               </apply>
             </apply>
             <apply>
               <csymbol cd="ambiguous">superscript</csymbol>
               <ci>𝑥</ci>
               <cn type="integer">2</cn>
             </apply>
           </apply>
           <apply>
             <ci>^</ci>
             <ci>𝑥</ci>
           </apply>
         </apply>
       </apply>
     </math>
step The
measure work
is calculated using W = \(\Delta E\) since the
measure force
changes. To find the cumulative
measure work
done on the
tangible entity object
, integrate over all positions between \(\infty\) and \(r\) \begin{equation} W = \int_{\infty}^r \vec{F}\cdot d\vec{r} \label{eq:work as function of force} \end{equation}
step Substituting the
measure gravitational force
into Eq. \ref{eq:work as function of force},
expression as Content MathML \begin{equation} W = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx \end{equation} 
<?xml version="1.0" encoding="UTF-8"?>
   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="W=\int_{\infty}^{r}\frac{-Gm_{1}m_{2}}{x^{2}}dx" display="block">
     <apply>
       <eq/>
       <ci>𝑊</ci>
       <apply>
         <apply>
           <csymbol cd="ambiguous">superscript</csymbol>
           <apply>
             <csymbol cd="ambiguous">subscript</csymbol>
             <int/>
             <infinity/>
           </apply>
           <ci>𝑟</ci>
         </apply>
         <apply>
           <times/>
           <apply>
             <divide/>
             <apply>
               <minus/>
               <apply>
                 <times/>
                 <ci>𝐺</ci>
                 <apply>
                   <csymbol cd="ambiguous">subscript</csymbol>
                   <ci>𝑚</ci>
                   <cn type="integer">1</cn>
                 </apply>
                 <apply>
                   <csymbol cd="ambiguous">subscript</csymbol>
                   <ci>𝑚</ci>
                   <cn type="integer">2</cn>
                 </apply>
               </apply>
             </apply>
             <apply>
               <csymbol cd="ambiguous">superscript</csymbol>
               <ci>𝑥</ci>
               <cn type="integer">2</cn>
             </apply>
           </apply>
           <apply>
             <csymbol cd="latexml">differential-d</csymbol>
             <ci>𝑥</ci>
           </apply>
         </apply>
       </apply>
     </apply>
   </math>
step Factor out the constants,
expression as Content MathML \begin{equation} W = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx \end{equation} 
<?xml version="1.0" encoding="UTF-8"?>
      <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="W=-Gm_{1}m_{2}\int_{\infty}^{r}\frac{1}{x^{2}}dx" display="block">
        <apply>
          <eq/>
          <ci>𝑊</ci>
          <apply>
            <minus/>
            <apply>
              <times/>
              <ci>𝐺</ci>
              <apply>
                <csymbol cd="ambiguous">subscript</csymbol>
                <ci>𝑚</ci>
                <cn type="integer">1</cn>
              </apply>
              <apply>
                <csymbol cd="ambiguous">subscript</csymbol>
                <ci>𝑚</ci>
                <cn type="integer">2</cn>
              </apply>
              <apply>
                <apply>
                  <csymbol cd="ambiguous">superscript</csymbol>
                  <apply>
                    <csymbol cd="ambiguous">subscript</csymbol>
                    <int/>
                    <infinity/>
                  </apply>
                  <ci>𝑟</ci>
                </apply>
                <apply>
                  <times/>
                  <apply>
                    <divide/>
                    <cn type="integer">1</cn>
                    <apply>
                      <csymbol cd="ambiguous">superscript</csymbol>
                      <ci>𝑥</ci>
                      <cn type="integer">2</cn>
                    </apply>
                  </apply>
                  <apply>
                    <csymbol cd="latexml">differential-d</csymbol>
                    <ci>𝑥</ci>
                  </apply>
                </apply>
              </apply>
            </apply>
          </apply>
        </apply>
      </math>
step which leads to
expression as Content MathML \begin{equation} W = \frac{G m_1 m_2}{r} \end{equation} 
<?xml version="1.0" encoding="UTF-8"?>
  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="W=\frac{Gm_{1}m_{2}}{r}" display="block">
    <apply>
      <eq/>
      <ci>𝑊</ci>
      <apply>
        <divide/>
        <apply>
          <times/>
          <ci>𝐺</ci>
          <apply>
            <csymbol cd="ambiguous">subscript</csymbol>
            <ci>𝑚</ci>
            <cn type="integer">1</cn>
          </apply>
          <apply>
            <csymbol cd="ambiguous">subscript</csymbol>
            <ci>𝑚</ci>
            <cn type="integer">2</cn>
          </apply>
        </apply>
        <ci>𝑟</ci>
      </apply>
    </apply>
  </math>
step Another definition of
measure work
is that it is the change in
measure energy
for a system: \(W = \Delta E \) Because the
measure initial velocity
was
value zero
, the
measure work
here is
expression as Content MathML \begin{equation} W = \Delta KE \end{equation} 
<?xml version="1.0" encoding="UTF-8"?>
       <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="W=\Delta KE" display="block">
         <apply>
           <eq/>
           <ci>𝑊</ci>
           <ci>ΔKE</ci>
         </apply>
       </math>
Thus we can combine the two definitions of
measure work
to get
expression as Content MathML \begin{equation} W = \frac{1}{2} m_1 v^2 = \frac{G m_1 m_2}{r} \end{equation} 
<?xml version="1.0" encoding="UTF-8"?>
   <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="W=\frac{1}{2}m_{1}v^{2}=\frac{Gm_{1}m_{2}}{r}" display="block">
     <apply>
       <and/>
       <apply>
         <eq/>
         <ci>𝑊</ci>
         <apply>
           <times/>
           <apply>
             <divide/>
             <cn type="integer">1</cn>
             <cn type="integer">2</cn>
           </apply>
           <apply>
             <csymbol cd="ambiguous">subscript</csymbol>
             <ci>𝑚</ci>
             <cn type="integer">1</cn>
           </apply>
           <apply>
             <csymbol cd="ambiguous">superscript</csymbol>
             <ci>𝑣</ci>
             <cn type="integer">2</cn>
           </apply>
         </apply>
       </apply>
       <apply>
         <eq/>
         <share href="#Ex1.m1.sh1"/>
         <apply>
           <divide/>
           <apply>
             <times/>
             <ci>𝐺</ci>
             <apply>
               <csymbol cd="ambiguous">subscript</csymbol>
               <ci>𝑚</ci>
               <cn type="integer">1</cn>
             </apply>
             <apply>
               <csymbol cd="ambiguous">subscript</csymbol>
               <ci>𝑚</ci>
               <cn type="integer">2</cn>
             </apply>
           </apply>
           <ci>𝑟</ci>
         </apply>
       </apply>
     </apply>
   </math>
step The \(m_1\) cancels, leaving
expression as Content MathML \begin{equation} v(r) = \sqrt{\frac{2Gm_2}{r}} \end{equation} 
<?xml version="1.0" encoding="UTF-8"?>
  <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v=\sqrt{\frac{2Gm_{2}}{r}}" display="block">
    <apply>
      <eq/>
      <ci>𝑣</ci>
      <apply>
        <root/>
        <apply>
          <divide/>
          <apply>
            <times/>
            <cn type="integer">2</cn>
            <ci>𝐺</ci>
            <apply>
              <csymbol cd="ambiguous">subscript</csymbol>
              <ci>𝑚</ci>
              <cn type="integer">2</cn>
            </apply>
          </apply>
          <ci>𝑟</ci>
        </apply>
      </apply>
    </apply>
  </math>