Calculate Earth's escape velocity using different methods

[Physicsdudee]

Homework Statement

We were asked to calculate earth's escape velocity using methods 1) Newton law of motion, 2) Conservation of energy, 3) Lagrange method.
As to conservation of energy, that is clear, and I got the result (ca. 11.19km/s).
However for 1) and 3) I have no Idea, though I might have a start for 3).
Help would be appreciated.
Thanks

Relevant Equations

F=m*a, Ekin+Epot=const

 

[PeroK]

For 1) you could use Newton's laws to generate a differential equation and apply the relevant boundary conditions.

I must confess I'm not sure what is meant by 3). You can use the Lagrangian to generate either Newton's laws or the conservation of energy (which follows from time invariance of the Lagrangian).

 

[vela]

Note that the equation
$$m\dot v + \frac{GMm}{r^2}=0$$ is just $F=ma$. You can't integrate it the way you did since $r$ is a function of $t$.

How are $r$ and $v$ related?

 

[Physicsdudee]

Hmm I mean I know that r and v are related in a way that dr/dt=v and vice versa with the integral.
I could replace the v' in this differential equation you posted with r''.
That would give me a second order differential equation in terms of r, however, if I solve for r, I get a function r(t). How do I get a constant velocity out of this? By differentiating I get a function v(t), but I have no initial condotins to be able to compute v_escape.

 

[PeroK]

Hmm I mean I know that r and v are related in a way that dr/dt=v and vice versa with the integral.
I could replace the v' in this differential equation you posted with r''.
That would give me a second order differential equation in terms of r, however, if I solve for r, I get a function r(t). How do I get a constant velocity out of this? By differentiating I get a function v(t), but I have no initial condotins to be able to compute v_escape.

I'm not sure what you are trying to do. Escape velocity is an initial condition at some initial $r_0$ in order to ensure that $v \rightarrow 0$ as $r \rightarrow \infty$.

 

[Physicsdudee]

Hmm yeah that makes sense. Tbh I don't really get the question we were given either. It said that we must calculate escape velocity using the Lagrange equation, that's it. But yeah, I will try to find a way, maybe some idea pops up.

 

[vela]

Hmm I mean I know that r and v are related in a way that dr/dt=v and vice versa with the integral.
I could replace the v' in this differential equation you posted with r''.
That would give me a second order differential equation in terms of r, however, if I solve for r, I get a function r(t). How do I get a constant velocity out of this? By differentiating I get a function v(t), but I have no initial conditions to be able to compute v_escape.

The usual trick is to multiply both sides of the equation by $\dot r$, which makes it easy to integrate. You should recognize the resulting equation.

 

[Physicsdudee]

The usual trick is to multiply both sides of the equation by $\dot r$, which makes it easy to integrate. You should recognize the resulting equation.

Thank you for the hint, I had totally forgotten about that trick. I think that made it work:)

 

[berkeman]

As an aside comment @Physicsdudee -- please check out the "LaTeX Guide" link below the Edit window. Posting math equations in LaTeX makes them *much* more legible. Thanks.