Velocity needed to escape gravitational potential

[Fibo112]

I am solving a problem where I need to decide if an asteroids velocity is high enough to escape the planets gravitational pull. The way I did it was use conservation of energy and angular momentum to find an expression for the radial velocity and show that it remains positive as r tends to infinity. In the solution all that was done was argue that the sum of potential and kinetic energy is positive. I can see how this would work if v0 is pointing radially outward but I don't see why is holds generally. Does it and if so why.

 

[russ_watters]

At infinity, what direction does the [other end] of the velocity vector point?

 

[Fibo112]

The velocity will point radially outward at infinity

 

[Dale]

The way I did it was use conservation of energy and angular momentum to find an expression for the radial velocity and show that it remains positive as r tends to infinity.

That is a good approach. You can probably use the same approach to find an expression for the angular velocity. What does it tend to?

 

[Fibo112]

Well the angular velocity will go to zero in any case where r goes to infinity

 

[jbriggs444]

That is a good approach. You can probably use the same approach to find an expression for the angular velocity. What does it tend to?

An angular velocity that tends to zero is not sufficient to guarantee that the tangential velocity also tends to zero. It may be better to reason based on conservation of angular momentum that the tangential velocity must tend to zero.

Further, an angular velocity that tends to zero is not sufficient to guarantee the existence of a limiting direction of travel.

But then I am not completely certain why @Dale is asking you to consider angular velocity in the first place.