A rocket satellite travels in a circular orbit of radius $\displaystyle r_0$. The rocket motor suddenly increases the rocket velocity by 8 % in its direction of motion. What is the apogee distance of the new orbit?
It is known that the tangential speed of a circular orbit is given by $\displaystyle v_C=\sqrt{\frac\mu r}$, an elliptical orbit by $\displaystyle v_E=\sqrt{\mu (\frac2r-\frac1a)}$ where a is the apogee of the orbit. If your rocket instantaneously increases its tangential speed by 8% then this point on the orbit will become the perigee of the new elliptical orbit, and it will pass through this point on every cycle. Therefore the ratio of its current elliptical orbit speed and its past circular orbit speed will always be a constant 1.08, i.e. $\displaystyle \frac{v_E}{v_C}=1.08$ AT THAT POINT. Solving this equation, $\displaystyle \frac a r =1.1996$ . Since this point represents the perigee of the orbit (r=p), then $\displaystyle \frac a p =1.1996$ . so the answer to your question is that the apogee of the new orbit will increase by 20% over the old constant circular radius.