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In this page the relation between "measure" and "symbol" pairs is illustrated visually. (The section, word, and expression tags have been suppressed for readability.) The underlying data structure for these relations could be a bipartite graph.

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

Suppose an object starts an infinite distance from a moon and is dropped, falling towards the moon due to gravitational acceleration. What is the speed of the object when it is

measure: r distance
symbol: distance \(r\)
from the moon?
Figure 1: small mass falling towards a moon from initial position at infinity.

The initial condition is \begin{equation} v(x=\infty) =0 \label{eq:initial_velocity} \end{equation} The force acting on the object is \begin{equation} \vec{F} = \frac{-G m_1 m_2}{x^2} \hat{x} \label{eq:gravitational force} \end{equation} The

measure: W work
is calculated using
symbol: work \(W\)
= \(\Delta E\) since the force changes. To find the cumulative work done on the 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 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: W work
is that it is the
measure: \(\Delta E \) change in energy
for a system:
symbol: work \(W\)
symbol \(\Delta E \)
Because the initial velocity was zero, the work here is \begin{equation} W = \Delta KE \end{equation} Thus we can combine the two definitions of 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}