## Relationaship Between Radius of Orbit of Planet and Orbital Speed

As the orbital radius of a planet increases, the distance the planet has to travel to complete one orbit increases, but so does the period of the orbit. To find the relationship between orbital speed and radius of orbit, equate the gravitational force to the centripetal force.\[\frac{GM_{Sun} m_{Earth}}{r^2}=\frac{m_{Earth}v^2}{r}\]

Hence

\[v=\sqrt{\frac{GM}{r}}\]

As the radius of the orbit increases the speed of the planet decreases. The kinetic energy decreases, but because the total energy (kinetic plus gravitational potential energy is negative and equal to half the gravitational potential energy

\[GPE= - \frac{GM_{Sun} m_{Earth}}{r}\]

) the total energy of the planet will increase.