Escape Velocity Calculations for Multiple Masses

[dynawics]

Hello,

I had some questions on escape velocity. The only methods that I have seen on escape velocity involve the assumption that the mass which is "being escaped from" remains at rest (at a fixed point in space). For example the method of determination of escape velocity using Ki+Ui=Kf+Uf seems to be useful only if the first mass is assumed to be at a fixed point in space. If there are other methods of determining escape velocity which account for the acceleration of the "mass which is escaped from" towards the "mass which is escaping" please point me in that direction.

I was also wondering if anyone knew how to calculate the escape velocity for a single mass in a system of three or more different masses at different points in space?

Thank you.

 

[Simon Bridge]

Welcome to PF;
If the escape velocity of body mass m from body mass M is v, then the escape velocity for body mass M from body mass m is also v.

i.e. you have to throw a ball at 11.2kmps straight up for it to escape the Earth ... this means you have to throw the Earth at 11.2kmps straight down for the Earth to escape the ball.

The most general calculation is Newton's law of gravitation with conservation of energy, and momentum. It's pretty much the same calculation except you are using a different place for your axes.

If you have two masses m with centers initially a distance r from each other, and we want to know how fast to separate them so they come to rest "at the edge of the Universe" ... the escape velocity calculation will give you the separation speed: the rate the distance between them must initially grow. If each have the same speed in opposite directions, then they need to have half the escape speed wrt their center of mass.

For escape velocity to be an issue, though, one mass is usually very much bigger than the other.

If there is more than one mass it is still the same calculation - you just have to add the contrbutions from each mass.

Some care should be taken though ... the escape velocity is worked out by making the initial kinetic energy equal to the gravitational potential energy for the position. This does not guarantee that the object will escape (it could, for eg, collide with one of the masses). Similarly, not having escape velocity does not mean that the object will not escape. A constant velocity of 1m/s upwards from the surface of the Earth will still get you anywhere you want to go - you just have to supply energy throughout the trip.