Question

At two different points in its orbit, a comet has velocities v1 and v2. If: V1and V2 are orthogonal and |V1] = 2|V2|,what is the smallest possible eccentricity of the orbit? 

Answer 1

Velocity V1 when it is perpendicular to the semi-minor axis:

$$v_1=\sqrt{\mu\bigg(\frac{2}{b}-\frac1a\bigg)}=2v_2.$$

For V2:

$$v_1=2v_2,\\\space\\ 2\sqrt{\frac \mu a}=\sqrt{\mu\bigg(\frac{2}{b}-\frac1a\bigg)},\\\space\\ 2\sqrt{\frac \mu a}=\sqrt{\mu\frac{2a-b}{ba}},\\\space\\ 4=\frac{2a-b}{b}=\frac{2a}{b}-1,\\\space\\ \frac52=\frac ab,\\\space\\ e=\sqrt{1-\frac{b^2}{a^2}}=\sqrt{1-\frac{2^2}{5^2}}=0.92.$$