Two identical planets, each with a mass of 10 24 kg, orbit around the midpoint between the two...

Question:

Two identical planets, each with a mass of {eq}10^{24} kg {/eq}, orbit around the midpoint between the two planets. If the distance between the two planets is {eq}2 \times 10^8 m {/eq} (as measured from their centers), how fast is each of the planets moving in their orbit? (EXPLAIN STEP BY STEP)

Orbital Velocity

Orbital velocity is defined as the velocity required by a heavenly body to orbit other massive body. If the mass of the massive body is M ans orbital radius is r then orbital velocity is given by

{eq}\begin{align} v = \sqrt{\frac{GM}{r}} \end{align} {/eq}

Let us recap important information from the question

• Mass of each planet {eq}m = 10^{24} \ kg {/eq}
• Distance between two planet {eq}d = 2 \times 10^8 \ m {/eq}

Here we will assume that both the planet are orbiting the center of mass of the two particle system and whole mass is concentrated at the center of mass i.e. mid point as both the particle are identical. Now total mass on the center of mass

• Mass {eq}M = m+ m = 2m = 2\times 10^{24} \ kg {/eq}
• Distance of each planet from midpoint i.e. orbital radius {eq}r = d/2 = 10^8 \ m {/eq}

Now orbital velocity is given by

{eq}\begin{align} v = \sqrt{\frac{GM}{r}} \end{align} {/eq}

{eq}\begin{align} v & = \sqrt{\frac{6.67 \times 10^{-11} \ N.m^2.kg^{-2} \times 2 \times 10^{24} \ kg }{10^{8} \ m}} \\ & = 1154.99 \ m/s \end{align} {/eq}

{eq}\begin{align} v & \approx 1155 \ m/s \end{align} {/eq}