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  1. Now earth is revolving around the sun with velocity of 29784.5029m/s and its velocity can be calculated from $v=\sqrt{\frac{GM}{r}}$ . How about earth's velocity suddenly changes into 25000m/s , 35000m/s or even some values? If the earth's velocity immediately changes, how can I predict its orbit shape?

Moreover,

  1. Orbit trajectory of the earth with less than 29784.5029 of revolving velocity.. Im researching for the earth's orbital change after comet crash in highschool and I got velocity less than the earth's current one without direction change of earth from calculating (earth's direction didn't change because its mass is so much bigger than comet's mass that from law of conservation of momentum, i got this result.) I don't know how to analyse earth's motion after collision, maybe it falls into the sun or orbits with elliptic orbit.

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  • $\begingroup$ feel free to comment/answer this! I wanna know this ㅠㅜ $\endgroup$
    – Junyeong
    Feb 1 at 6:00
  • $\begingroup$ You need to think about energy as well as momentum. A useful concept here is the specific orbital energy. Some parts of that article may be a bit confusing or too technical, but I think you'll find it helpful. $\endgroup$
    – PM 2Ring
    Feb 1 at 7:26
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Let me address your second point first; all Newtonian/Kepler orbits are elliptical; that means that the earth is currently in an elliptical orbit and will remain in one if a comet hits it (if the comet is not so large or so fast to cause the earth to cease orbiting). But I assume in your class; you are working with the circular orbit approximation; so I will use that in answering your question.

You state correctly that $v=\sqrt{\frac{GM}{R}}$ and so obviously $G$ and $M$ don't change, therefore $R$ must change to maintain the equality. Specifically. $\frac{v_1^2}{v_2^2}R_1=R_2$. Again this is using spherical orbit approximations; in reality there are more involved differential equations that require you to specify "how" the velocity changes.

Now if a comet of mass $m_c$ and velocity $v_c$ where to crash into earth, you would have to run through the conservation of momentum calculations to find the new velocity; and plug it into the equations above. But there is a point of importance here which is "at what angle does the commet hit earth"? You see if the comet hits the earth in the same direction as its velocity; then the conservation of momentum can be done in only one direction and we are good, however if it has an angle; or is even perpendicular to the earth's velocity; then it will "squish" the orbit into an ellipse (note: since in reality we are already in an ellipse; this would just change the 'ovalness' of that ellipse).

These types of questions have many layers of depth and intricacy and you just have to decide where you want to end, if the comet is moving really fast and really heavy it can force the earth to reach its escape velocity and leave orbit; or it can force the earth to crash into the sun. Moreover because of the conservation of angular momentum of the earth's rotation about its own axis; this comet would change the tilt of the earth and in turn affect our seasons (I believe this is why Uranus has a tilt of $82^o$).

Something that might be of interest to you is looking up why orbits are elliptical; and we can explain it using Newton's Laws which I'm sure you are familiar with. This is an absolutely beautiful video by Grant Sanderson which explains it better than I ever could: https://www.youtube.com/watch?v=xdIjYBtnvZU&t=453s

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  • $\begingroup$ Really thank you for your answer. And exactly i already calculated earth's velocity after crash, which is by one of the popular comet, halley's . I have drawn earth's and halley's orbits from making elliptical equation using Geogebra and calculated it. let halley's comet crash into earth's ocean and set restitution coefficient as e=0, then it collides as perfect inelastic collision and the sum mass of earth and halley's gets velocity calculated as 29784.5m/s after crash . i got this value but i cant know how they(the sum mass of earth and halley's) would move like? $\endgroup$
    – Junyeong
    Feb 1 at 10:11

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