Can the escape velocity be achieved at any angle?

[nhmllr]

So I looked at a neat derivation for what the minimun escape velocity is, and It was pretty clever. Because this is conserved:
KE + PE = 1/2 *mv^2 + -GMm/r
You can find what the velocity would have to be to get to infinity with zero velocity
1/2 *mv_esc^2 + -GMm/r = 1/2 *m(0)^2 + -GMm/(infinity) = 0
1/2 *[STRIKE]m[/STRIKE]v_esc^2 = GM[STRIKE]m[/STRIKE]/r
v_esc = sqrt(2GM/r)
However, there is no "angle" term included in this. Does this mean that it can travel at this velocity at any angle and it will escape? What if the velocity vector is pointed right at the thing it's orbiting?

Thanks

 

[Yuqing]

I believe that velocity has to be directed radially outwards. The escape velocity is defined as the velocity required for an object to "get to infinity" with zero remnant velocity. If the velocity is not radially, then that cannot happen from the conservation of angular momentum.

 

[cjl]

I believe that velocity has to be directed radially outwards. The escape velocity is defined as the velocity required for an object to "get to infinity" with zero remnant velocity. If the velocity is not radially, then that cannot happen from the conservation of angular momentum.

Nope - it works just fine regardless of angle. Keep in mind that from a conservation of angular momentum point of view, v can still go to zero (as r -> inf).

 

[nhmllr]

Nope - it works just fine regardless of angle. Keep in mind that from a conservation of angular momentum point of view, v can still go to zero (as r -> inf).

But mathematically, if you pointed the velocity vector straight at the object with mass M, the force would be infinite when r = 0, and the object would be stationary. Right? Or does the velocity also grow to be near infinite, kind of making them "cancel out?"

 

[Yuqing]

Nope - it works just fine regardless of angle. Keep in mind that from a conservation of angular momentum point of view, v can still go to zero (as r -> inf).

Ah yes, forgot about the fact that r -> inf.

@nhmllr It doesn't really make physical sense to point the velocity towards the object. Escape velocity is defined in terms of the object "escaping" the planet.