Calculate Earth's escape velocity using different methods

In summary, this conversation discusses how to generate a differential equation and apply the relevant boundary conditions. Newton's laws can be generated using the Lagrangian, and the conservation of energy can be derived from time invariance of the Lagrangian.
  • #1
Physicsdudee
14
2
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
try.PNG
 
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  • #2
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).
 
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  • #3
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?
 
  • #4
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.
 
  • #5
Physicsdudee said:
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##.
 
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  • #6
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.
 
  • #7
Physicsdudee said:
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.
 
  • #8
vela said:
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:)

try.PNG
 
  • #9
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.
 
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FAQ: Calculate Earth's escape velocity using different methods

1. What is escape velocity?

Escape velocity is the minimum speed that an object needs to achieve to escape the gravitational pull of a celestial body, such as Earth.

2. How is Earth's escape velocity calculated?

Earth's escape velocity can be calculated using the formula v = √(2GM/r), where G is the gravitational constant, M is the mass of the Earth, and r is the distance from the center of the Earth to the object.

3. What are the different methods for calculating Earth's escape velocity?

There are two main methods for calculating Earth's escape velocity: using the formula mentioned above and using the energy conservation equation, KE + PE = 0, where KE is the object's kinetic energy and PE is its potential energy.

4. How does Earth's escape velocity vary with altitude?

As an object moves further away from the Earth's surface, the escape velocity decreases. This is because the force of gravity weakens with distance, making it easier for the object to escape the Earth's pull.

5. What is the significance of calculating Earth's escape velocity?

Calculating Earth's escape velocity is important for space travel and understanding the dynamics of celestial bodies. It helps determine the minimum speed needed for spacecraft to leave Earth's orbit and explore other planets or objects in space.

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