Fall 2005 1 Mech

A meteorite of mass M1 is incident with the relative velocity v0 and impact parameter s on a planet of radius R and mass M2. Determine the largest s for the collision to occur assuming that the two bodies attract each other according to Newton’s law of gravitation.

Hint: Use polar coordinates.

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Answer

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(1)
\begin{align} \mbox{const} = H =\frac{p_r^2}{2m}+\frac{p_{\theta}^2}{2mr^2}-\frac{G M_1 M_2}{r} \end{align}

where $m = \frac{M_1 M_2}{M_1+M_2}$ and $p_{\theta}=mv_0 s$

set $p_r = 0$ at r=R.

(2)
\begin{align} \frac{mv_0^2}{2}+\frac{GM_1M_2}{R}=\frac{m^2v_0^2s^2}{2mR^2} \end{align}
(3)
\begin{align} s^2 = R^2 \left[1+\frac{G M_1 M_2}{R}\frac{1}{\frac{1}{2}\left( \frac{M_1M_2}{M_1+M_2}\right) v_0^2 } \right] \end{align}
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