Why escape velocity is independent of angle of projection

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Escape velocity is independent of the angle of projection because it relies on the conversion of kinetic energy to potential energy, which is not direction-dependent. This principle is rooted in the conservative nature of gravitational fields, where the work done does not vary with the path taken. In an ideal scenario without atmospheric resistance, any launch angle will ultimately lead to a trajectory that approaches a straight line away from the planet. Geometrically, since escape velocity aims to reach infinity, all paths converge to a direction that is effectively away from the gravitational source. Thus, the angle of projection does not influence the required escape velocity.
Himal kharel
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can someone explain physically why escape velocity is independent of angle of projection.
 
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Because it is based on transfer of kinetic energy to potential energy which is independent of direction.
 
A short answer can be: because the gravitational field is conservative (work does not depend on path). I suppose by "angle of projection" you mean some angle between the initial velocity and some direction related to the planet.

Of course, this is true in the ideal case, without atmosphere.

Edit. Someone answered already, sorry.
 
Could it be looked at geometrically? Since escape velocity takes you to infinity, any path will end up forming a ray arbitrarily close to directly away from the object.
 

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