Velocity of an satellite in an elliptical orbit

The semi-major axis equals to $\dfrac{a+b}{2}$ where $a<b$.

By vis-viva equation,

$$v^2=2GM\left( \frac{1}{r}-\frac{1}{a+b} \right)$$

At perigee ($r=a$),

$$v_a=\sqrt{\frac{2GMb}{a(a+b)}}$$

At apogee ($r=b$),

$$v_b=\sqrt{\frac{2GMa}{b(a+b)}}$$

Both total energy and angular momentum are conserved.

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