# That was a close call!

A heavy body of mass $$m$$ (shown in grey) approaching Earth with initial velocity $$v_0$$ in the absence of interaction has an aiming parameter $$a$$. At the instant of closest approach to Earth, its distance from the surface of Earth is given by

$b = \dfrac{GM_e}{{(v_0)}^{y}} \left[ \sqrt{x + {\left( \dfrac{a \ {(v_0)}^{y}}{z \ GM_e} \right)}^2} -1 \right] - R_e,$

where $$x , y,$$ and $$\ z$$ are positive constants.

Find $$x + y + z$$.

Details and Assumptions:

• Assume there are no other external bodies other than Earth and the heavy body. The only mutual force of gravitation that exists is between them.
• Neglect the size of the heavy body, i.e. treat it as a particle of mass $$m$$.
• $$G$$ denotes the universal gravitational constant, i.e. $$G = 6.67 \times 10^{-11} \text{ N m}^2 \text{ kg}^{-2}$$.
• $$M_e$$ denotes the mass of Earth, i.e. $$M_e = 5.972 \times 10^{24} \text{ kg}$$.
• $$R_e$$ denotes the radius of Earth, i.e. $$R_e = 6371 \text{ km}$$.
• Assume that rotation and revolutions of Earth are absent for this problem.
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