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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