In previous physics briefs by Jason Jarvis, found at https://JarvisLabs.com/Jupiter, Jarvis illustrates and explains the origin of rocky or icy terrestrial planets from within the gas giants. Jarvis hypothesizes that the Earth and Moon orbiting pair originated in the Jupiter system and are very close astronomical neighbors of Jupiter's Galileon moons. This physics brief titled "Back of the Hand Momentum/Velocity Calculations in Support of Jarvis Earth & Moon Origination Theory" was realized when performing and contemplating thought experiments regarding this planetary formation theory. This argument and observation holds to basic conservation of momentum laws as we understand and accept them.
Another Physics Brief by Jason Jarvis  released to the public on or around March 28^{th}, 2018
In its most basic form the conservation of momentum implies that the total momentum in a system, which is not acted upon by external forces ( in a perfectly closed system ), remains constant over time. (For reference see https://en.wikipedia.org/wiki/Momentum). Newton's laws of motion are credited with this implication.
A simple formula illustrating this concept for the interaction of 2 bodies, M_{1} and M_{2}, could be written as:
M_{1}V_{11} + M_{2}V_{12} ≈ M_{1}V_{21} + M_{2}V_{22}
where,
V_{11} = initial velocity of M_{1} at t_{1}
V_{12} = initial velocity of M_{2} at t_{1}
and,
V_{21} = velocity of M_{1} at later time (t_{2})
V_{22 }= velocity of M_{2} at later time (t_{2})
In 3dimensional space velocity can be modeled with vector math indicating direction and relative speeds within a common reference frame.
For the purpose of this brief we will note that all solar system bodies are affected by the Sun's gravity as well as each and every one of its neighbors. However, for discussion purposes we assume nothing specific other than total masses for the Earth and Moon are constant over billions of years and we neglect the micro effects of drag on bodies through space and the reflexivity or impact absorption between them. Thus back of the hand, gross, conservation of momentum concepts are discussed here.
Let's now consider what we know about the velocities of Earth, Moon, Jupiter and the Galilean moons. The following information was obtained from NASA's Solar System Exploration web site.
Planetary Body  Orbit Velocity around the Sun  Moon Orbit Velocity around its home planet 
Earth 
107,218 km/h


Moon  107,218 km/h* 
3,680.5 km/h

Jupiter 
47,002 km/h


Io 
62,423.1 km/h


Europa 
49,476.1 km/h


Ganymede 
39,165.6 km/h


Callisto  29,531.6 km/h 
The Earth and Moon orbit as a pair around the Sun at an estimated velocity of ~107,218 km/h. Earth's moon orbits around the Earth at ~3,680 km/h. The Jupiter system orbits around the Sun at a velocity of ~47,002 km/h. Io, Europa, Ganymede, and Callisto orbit at ~62,423 km/h, ~49,476 km/h, 39,165 km/h, and ~29,531 km/h respectively as shown in the table above.
According to the Jarvis planetary formation and origin of Earth theory, we postulate that the Earth and Moon were both a Body N which was formed internal to Jupiter (ie, what is going on under the great red spot). They then exited Jupiter's upper atmosphere and orbited Jupiter much the same as the current Galilean moons. Then at some point, billions of years prior to now, the ancient Earth and Moon bodies which were in orbit around Jupiter collided.
This collission between ancient Earth and Moon bodies was most probably the gravitational shove that allowed this new entanlged orbiting pair to break away from Jupiter's close orbit. As the Earth and Moon gradually moved further from Jupiter's gravity their major velocities with respect to the sun stayed in tact, due to conservation of momentum, and they settled into an elliptical orbit around the Sun.
Consider now the average of all orbit velocities for the Galilean moons about Jupiter.
Average of ( 62,423.1 + 49,476.1 + 39,165.6 + 29,531.6 ) = 45,149.1 km/h  Average orbit velocity of Galilean moons.
For rough back of the hand physics now let's add this velocity to the Jupiter orbit velocity to get an idea of how fast an escaped body from the Jupiter system might be going relative to the Sun.
Addition of ( 45,149.1 + 47,002 ) ≈ 92,151 km/h
Statistically speaking the Earth/Moon velocity of ~107 km/h and the orbit velocity of Mars at ~86,677 km/h are close enough to this averaged velocity of ~92 km/h that you can pursue correction factors, other assumptions, or weighted averages over time to justify this as a viable theory.
The most likely scneario for escape from Jupiter of its moons would be in the direction of the Sun as this is when its velocity is highest towards the Sun and most break away potential is. That is, with each moon orbit, whether the Earth's moon or Jupiter's moon the Sun tugs it slightly further from its home planet.
It is probable that other unfound planetary bodies in our solar system were the result of this momentum process and that other bodies all together escaped our system to transit interstellar space.
Thanks for your interest in this topic. Please see other momentum gravity papers on the Jarvis Labs site for a complete picture of this theory, planetary formation, and more.
Continuing with Gravity and Mass Momentum, on to Venus
Gravity, mass momentum, and the Super Rotation of Venus's Atmosphere