A small meteor approaches a planet on a hyperbolic path as shown. When it is at a large distance, it has a velocity v∞ and impact parameter b = 8000 km. The escape speed from the planet is 6 km/s. What should be v∞ (in km/s) so that the meteor just grazes the planet. The radius of planet is 6400 km.

Engineering

Physics

Gravitational Potential Energy

Angular Momentum and Conservation of Angular Momentum

Question

A small meteor approaches a planet on a hyperbolic path as shown. When it is at a large distance, it has a velocity v_∞ and impact parameter b = 8000 km. The escape speed from the planet is 6 km/s. What should be v_∞ (in km/s) so that the meteor just grazes the planet. The radius of planet is 6400 km.

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L = const.
mbV_∞ = mV₁ × R

$- \frac{GMm}{\infty} + \frac{1}{2} {mV}_{\infty}^{2} (\frac{b^{2}}{R^{2}} - 1)$

$V_{\infty} = \sqrt{\frac{2 GM}{R}} \times \frac{R}{\sqrt{b^{2} - R^{2}}} = 6 \times \frac{6400}{\sqrt{800^{2} - 6400^{2}}}$

$= 6 \times \frac{6400}{800 \sqrt{100 - 64}} = 8 km / s$ .