As we all know Jupiter is a gaseous planet so what would happen if an Earth-sized object happened to encounter Jupiter?
or what would be the effect?
I like the answers, but I'm going to give a brief intuitive answer in addition.
Would Jupiter engulf the object and make it stay at Jupiter's core?
reading the other question, which I'm sure you read as well, here I think the safe answer is yes, Earth would be engulfed in Jupiter, though the impact spash would be impressive. I wouldn't want to guess as to exactly what would happen, but it would be, impressive.
Now if it was more of a glancing blow, large chunks of the Earth like planet could pass through and escape, but Jupiter is 10 times the diameter of Earth and quite solid with very high pressure inside. It's hard to see anything short of a Neutron Star passing through that with relative ease. The Jupiter mass rocky planet scenario wouldn't be so much a pass through as a blow up.
As pointed out, the Earth like planet simply approaching Jupiter would begin to break apart.
Would the object get fragmented as when two solid objects hit each other?
Fragmentation is slowed by gravity. When Earth was hit by Theia - a mars sized planet, at, not sure the velocity but lets say 15-25 KM per second, as it was likely a Trojan orbit, similar to Earth's orbit so the velocity was largely in line, and mostly driven by Earth's and Theia's combined gravitational pull towards each other. If you imagine 2 rocks smashing into each other at 20 times the speed of a bullet, the rocks would pulverize each other, and that's basically what happened with Earth and Theia, you got pulverized planets, a whole lot of heat and the Earth was sent a spinning, but you don't always get debris everywhere cause gravity is a big factor and that can keep a planet together.
This is an Earth-Theia simulation, a glancing blow (a direct hit and more mass would likely have been lost).
And this is probobly a pretty good estimate for an Earth like planet hitting Jupiter. Most of the debris would fall back to jupiter, a share, would have a good chance of forming either a ring (inside the Roche limit) or a moon (outside Jupiter's Roche limit) and a fair bit of matter would fly off and escape Jupiter's gravity given an impact that large. But as a general rule, the larger the planet being hit, the harder it is to blow material into space. Still, a collision this big, I think it would happen.
or what would be the effect?
The Earth, being 1/300th the mass of Jupiter, the crash would change Jupiter's orbit slightly but measurably, by a few tens of MPH and perhaps having a similar smallish effect on it's rotational velocity.
The collision could also be hot enough to make Jupiter glow, probobly enough to be clearly visible to the naked eye. How much - too hard for me to say, but perhaps it would look like a tiny sun but far outshining Venus, perhaps even, approaching the brightness of the full moon for a few hours. The glow might also change with Jupiter's rotation, which takes 10 hours, you could see it grow brighter and dimmer over a 10 hour period for a few days. It would be quite the show through a telescope, assuming it wouldn't be too bright, but you could watch the debris cloud take shape over several hours and perhaps form into a new moon in a few days.
Finally, another effect would likely be debris, even if it's mostly hydrogen, scattered around the solar system. If Jupiter was rocky, it would be far worse for us, as you'd likely get some enormous meteor showers in a few years and perhaps a few major meteor impacts (for example, if this happened on Mars, it would be quite bad for us), but Jupiter's more gaseous composition, I don't think that would be the case. The new dust clouds would be more curiosities than things to worry about.
It's worth pointing out that an earth sized object simply passing through the solar system could be quite problematic, not affecting the planets too much, but stirring up Kuiper belt objects or asteroids in the asteroid belt and sending them every which way, perhaps a few towards Earth - not a good situation for us, but that kind of new object passing through happens very rarely.
If you saw the video of the Shoemaker-Levy collision you will see that although that was a collection of smaller objects, there is very much a collision, rather than a gas engulfing a solid.
Once the object gets within Jupiter's Roche limit it will be torn to pieces, so you will have a hail of objects falling in and burning up through the atmosphere.
The Roche limit (pronounced /ʁoʃ/ in IPA, similar to the sound of rosh), sometimes referred to as the Roche radius, is the distance within which a celestial body, held together only by its own gravity, will disintegrate due to a second celestial body's tidal forces exceeding the first body's gravitational self-attraction.
This distance for Earth would be about 16.35 Earth radii, or 104,200 km.
Large enough chunks may survive and 'float' at a depth that matches their density - this may not be at the core...
Depending on what you mean by encounter:
If the planet comes into Jupiter's gravity influence the trajectory will be changed significantly. If it no longer has the necessary velocity to escape then it may be captured and become a new moon of Jupiter. On a smaller scale, this is what likely happened around Mars with Phobos and Deimos.
If the Earth-sized planet would encounter Jupiter within it's Roche Limit it would be torn apart. It's possible then that the torn apart debris would settle into a more vibrant ring system than Jupiter's current. This is possibly what happened to one of Saturn's moons to form it's current ring system.
If the Earth-sized planet is to directly hit Jupiter, it would follow the same as above to begin; the planet would reach an altitude below the Roche Limit and be broken into smaller pieces. These pieces would be further broken down as the rubble moved closer, until the final pieces impacted with Jupiter. The pressure of the pieces moving through the Jovian atmosphere would likely cause a heatup and explosion, similar to objects entering Earth's atmosphere. The majority, if not the entirety of the objects would be vaporized as a result.
