Question

A satellite is revolving round the Earth with the orbital speed $v_0$. If suddenly its speed becomes $2v_0$, then it will

• A. Move in elliptical path
• B. Move in circular path
• C. Have parabolic escape
• D. Have hyperbolic escape

Answer 1

Answer: (D)

Have hyperbolic escape

Answer 2

For a satellite in a circular orbit of radius $r$ with speed $v_0$, the centripetal force is provided by the gravitational force: $\frac{mv_0^2}{r} = \frac{GmM}{r^2}$ This gives the orbital speed as $v_0 = \sqrt{\frac{GM}{r}}$.

The escape velocity $v_e$ from radius $r$ is when the total energy is zero: $\frac{1}{2}mv_e^2 = \frac{GmM}{r}$ $v_e = \sqrt{\frac{2GM}{r}} = \sqrt{2} \sqrt{\frac{GM}{r}} = \sqrt{2} v_0$

The new speed of the satellite is $v' = 2v_0$. Comparing the new speed with the escape velocity: $v' = 2v_0$ $v_e = \sqrt{2} v_0 \approx 1.414 v_0$

Since $v' = 2v_0 > v_e = \sqrt{2} v_0$, the satellite's speed is greater than the escape velocity. When a satellite's speed exceeds the escape velocity, it will escape Earth's gravitational pull, and its trajectory will be hyperbolic.