Question

Two planets that are very similar in all aspects to Jupiter are released from rest when they are 2.00×1012 m away from each other. What is the speed of the planets as they collide? The mass of Jupiter is 1.90×1027 kg and the radius is 7.00×107 m.

Answer 1

Mass of Jupiter = 1.8986 * 10^27 kg
Mean radius = 69,911 km = 6.9911 * 10^7 meters

The Universal Gravitational force = G * M1 * M2 ÷ d^2
d = distance from the center of mass of one planet to the center of mass of the other planet.

The Universal Gravitational force is the force which causes the 2 planets to accelerate toward each other. As the planets approach each other, d decreases, so, the force increases. When they collide, the surface of one is touching the surface of the other planet. At this point, the distance from the center of mass of one planet to the center of mass of the other planet = 2 * radius = 2 * 6.9911 * 10^7 = 1.39822 * 10^8 meters

The distance between the planets decreased from 6.00* 10^10 meters to 1.39822 * 10^8 meters.
The total distance that the planets moved = 6.00* 10^10 – 1.39822 * 10^8 = 5.986018 * 10^10 m

Gravitational Potential energy = -G * M1 * M2 ÷ d
Since the 2 masses are the same, Gravitational Potential energy = -G * M^2 ÷ d


As the planets move closer, some of their gravitational potential energy is converted into kinetic energy.

Initial GPE = -6.67 * 10^-11 * (1.8986 * 10^27)^2 ÷ 2 * 10^12 = -1.2 * 10^32
Final GPE = -6.67 * 10^-11 * (1.8986 * 10^27)^2 ÷ 1.39822 * 10^8 = -1.719559774 * 10^36
Initial GPE – Final GPE = -1.719559774 * 10^36 – -1.2 * 10^32

Initial GPE – Final GPE = 1.715552569 * 10^36
This gravitational energy was converted into the kinetic energy of the 2 planet.

So, the kinetic energy of each planet = ½ * 1.715552569 * 10^36 = 8.577762845 * 10^35

½ * 1.8986 * 10^27 * v^2 = 8.577762845 * 10^35

v^2 = 8.577762845 * 10^35 ÷ 1.8986 * 10^27
The velocity of each planet = 2.13 * 10^4 m/s