## Roadmap for Formal Mathematical Physics Content

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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 distance $$r$$ from the moon?

Figure 1: small mass falling towards a moon from initial position at infinity.

added step There are two ways to characterize work in the context of a system. The two ways are equivalent $$W_{\rm to\ system} = W_{\rm by\ system}$$
The initial conditions are $$v(x=\infty) =0 \label{eq:initial_velocity}$$
added step and $$U_g(r=\infty) = 0$$
The force acting on the object is $$\vec{F} = \frac{-G m_1 m_2}{x^2} \hat{x} \label{eq:gravitational force}$$ The work is calculated using W_{\rm to\ system} = $$\Delta E$$ since the force changes. To find the cumulative work done on the object, integrate over all positions between $$\infty$$ and $$r$$ $$W_{\rm to\ system} = \int_{\infty}^r \vec{F}\cdot d\vec{r} \label{eq:work as function of force}$$ Substituting the gravitational force into Eq. \ref{eq:work as function of force}, $$W_{\rm to\ system} = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx$$ Factor out the constants, $$W_{\rm to\ system} = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx$$
added step Integrate to get $$W_{\rm to\ system} = -G m_1 m_2 \left(\left.\frac{-1}{x}\right|^r_{\infty}\right)$$ Then expand $$W_{\rm to\ system} = -G m_1 m_2 \left(\frac{-1}{r} - \frac{-1}{\infty}\right)$$ Simplify to get
$$W = \frac{G m_1 m_2}{r}$$ Another definition of work is that it is the change in energy for a system: $$W_{\rm by\ system} = \Delta KE$$.
added step In this characterization $$\Delta PE = 0$$.
Because the initial velocity was zero, the work here is $$W_{\rm by\ system} = \frac{1}{2} m_1 v^2$$ Thus we can combine the two definitions of work to get $$\frac{1}{2} m_1 v^2 = \frac{G m_1 m_2}{r}$$ The $$m_1$$ cancels, leaving $$v(r) = \sqrt{\frac{2Gm_2}{r}}$$