This article solves the first part of the problem proposed in a Chinese article on my Zhihu account. The original article was posted at 2020-08-30 18:27 +0800.

The aliens intiated their attack to the earth! They shoot bullets with mass $m$ and speed $v$ from a far-awar planet. To defend, humans built a field $U=\alpha/r$ that can repel the bullets. What regions are safe?

Every possible trajectory of the bullet is parameterized by $b$, the impact parameter. The bullet has energy $E=\frac12mv^2$ and angular momentum $M=mvb$, which are conserved. According to the well-known results of Kepler problem, the trajectory is a hyperbola

\[-\frac pr=1+e\cos\varphi,\]


\[p:=\frac{M^2}{m\alpha},\quad e:=\sqrt{1+\frac{2EM^2}{m\alpha^2}}.\]

For convenience, denote the radius of the hyperbola as


then we can write the equation of the trajectory as


Rotate the trajectory so that the incident direction is always towards the positive $x$ direction:

\[\begin{equation} \label{eq: trajectory} 0=F\!\left(r,\varphi,b\right):=\frac{b^2}{ar}+1+\cos\varphi+\frac ba\sin\varphi. \end{equation}\]

To find the envelope of the family of trajectories, solve

\[0=\frac{\partial F}{\partial b}=\frac{2b}{ar}+a\sin\varphi,\]

and we have


Substitute back into Equation \ref{eq: trajectory}, and we have finally the equation of the envelope:


which is a parabola with the the semi-latus rectum being $4a$. Therefore, the safe regions are the interior of a circular paraboloid.