Roadmap for Formal Mathematical Physics Content


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

Suppose an

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


condition initial condition
is \begin{equation} v(x=\infty) =0 \label{eq:initial_velocity} \end{equation} The
measure force
acting on the
tangible entity object
is \begin{equation} \vec{F} = \frac{-G m_1 m_2}{x^2} \hat{x} \label{eq:gravitational force} \end{equation} 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} Substituting the
measure gravitational force
into Eq. \ref{eq:work as function of force}, \begin{equation} W = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx \end{equation} Factor out the constants, \begin{equation} W = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx \end{equation} which leads to \begin{equation} W = \frac{G m_1 m_2}{r} \end{equation} 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
value zero
, the
measure work
here is \begin{equation} W = \Delta KE \end{equation} Thus we can combine the two definitions of
measure work
to get \begin{equation} W = \frac{1}{2} m_1 v^2 = \frac{G m_1 m_2}{r} \end{equation} The \(m_1\) cancels, leaving \begin{equation} v(r) = \sqrt{\frac{2Gm_2}{r}} \end{equation}