Equation of motion of particle in gravitational field

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SUMMARY

The discussion focuses on deriving the equation of motion for a particle (m1) influenced by the gravitational field of another particle (m2). The key equations referenced include Newton's law of universal gravitation, F=Gm1m2/r^2, and Newton's second law, F=ma. The acceleration of m1 is expressed as a=-Gm2/(x2-x1)^2, leading to the second-order differential equation d^2x1/dt^2=-Gm2/(x2-x1)^2. The participants emphasize the need for an analytical solution due to the changing distance between the particles.

PREREQUISITES
  • Understanding of Newton's law of universal gravitation (F=Gm1m2/r^2)
  • Familiarity with Newton's second law of motion (F=ma)
  • Knowledge of differential equations and their applications in physics
  • Basic concepts of kinematics, particularly relating to motion under variable acceleration
NEXT STEPS
  • Study analytical solutions for second-order differential equations in physics
  • Learn about numerical methods for solving differential equations, such as the Runge-Kutta method
  • Explore gravitational interactions in multi-body systems using tools like MATLAB or Python
  • Investigate the implications of variable gravitational fields on particle motion
USEFUL FOR

Physics students, researchers in classical mechanics, and anyone interested in understanding gravitational interactions and motion equations in a multi-body context.

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Homework Statement


We have two particles, m1 and m2 at positions x1 and x2, and we want to come up with the equation of motion of particle m1 due to the gravitational field of m2 (position of m1 as a function of time).


Homework Equations


F=Gm1m2/r^2
F=ma


The Attempt at a Solution


m1a=-Gm1m2/r^2
a=-Gm2/(x2-x1)^2
d^2x1/dt^2=-Gm2/(x2-x1)^2

Now, I don't know how to solve this. Any help will by appreciated.
 
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im confused at what ur asking for but here we go it might give u a start

F= Gm1m2/r^2
F=ma

m1a=Gm1m2/r^2

a=Gm2/r^2

v/t=Gm2/r^2

v=Gm2t/r^2

then use Speed = dist/time

is that what u wanted?
 
Aaron321 said:
im confused at what ur asking for but here we go it might give u a start

F= Gm1m2/r^2
F=ma

m1a=Gm1m2/r^2

a=Gm2/r^2

v/t=Gm2/r^2

v=Gm2t/r^2

then use Speed = dist/time

is that what u wanted?
The accelaration changes since the distance between the particles changes, so I think this has to be solved analytically. I just don't know how to handle the final equation I gave, d^2x1/dt^2=-Gm2/(x2-x1)^2
 

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