Solution of equation of free fall of a particle in a gravitational force field

[kevinaltieri]

Hi everybody, I am trying to find an analytical solution of the equation of a falling particle of mass M in a gravitational field (described by constant K) when the gravity depends on distance, as in Newton classical theory(I simplified in one dimension). The equation could be M*d2 X(t)/ dt2 = -K * 1/x(t)**2. Initial condition X(t0)= X0 and initial speed V0. I tried to find a solution and a range of existence of this solution with tecniques of second order non linear differential equation with separation of variables, but I lost myself on the road. Could somebody help me. Thank you very much.

 

[jbsteele]

Unfortunately, there is no known analytical solution to this equation. However, you can use numerical methods such as Euler's method or Runge-Kutta techniques to obtain approximate solutions.

 

[tomcue]

Hello,

Thank you for sharing your work on finding an analytical solution for the equation of free fall in a gravitational force field. This is a complex problem and it is understandable that you may have faced challenges in finding a solution.

One approach to solving this equation is to use the principle of conservation of energy. This states that the total energy of a system remains constant, and in the case of a falling particle, this energy is a combination of its kinetic energy and potential energy due to gravity.

Using this principle, you can derive an equation for the velocity of the particle as a function of time, and then integrate it to find the position of the particle as a function of time. This will give you an analytical solution for the equation of free fall.

Additionally, you can also use numerical methods such as Euler's method or Runge-Kutta method to approximate the solution and find the range of existence of the solution.

I hope this helps guide you in finding a solution for your problem. Keep exploring and don't give up, as finding analytical solutions to complex equations can be a challenging but rewarding experience. Best of luck!