Why escape velocity is independent of angle of projection

[Himal kharel]

can someone explain physically why escape velocity is independent of angle of projection.

 

[HallsofIvy]

Because it is based on transfer of kinetic energy to potential energy which is independent of direction.

 

[nasu]

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.

 

[russ_watters]

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.

 

[PJC01]

Escape velocity is the minimum speed required for an object to escape the gravitational pull of a planet or other celestial body. It is independent of the angle of projection because it is solely determined by the mass of the celestial body and the distance from its center.

To understand this concept, let's imagine a scenario where an object is projected at a certain angle from the surface of a planet. The gravitational force acting on the object is always directed towards the center of the planet, regardless of the angle of projection. This means that no matter what angle the object is projected at, it will always experience the same gravitational force pulling it towards the planet.

Now, if we increase the speed of projection, the object will travel farther away from the planet before being pulled back by gravity. This means that the object will have a higher potential energy at a greater distance from the planet's center. As the object moves away from the planet, it will also experience a decrease in its kinetic energy due to the gravitational force slowing it down.

The key point here is that the total energy of the object (kinetic energy + potential energy) remains constant throughout its trajectory, regardless of the angle of projection. This is because energy is a conserved quantity, meaning it cannot be created or destroyed. Therefore, the object will reach a point where its kinetic energy is completely converted into potential energy and it will escape the gravitational pull of the planet.

This point, where the object has just enough energy to escape the planet's gravitational pull, is the escape velocity. As you can see, it is solely determined by the mass and distance of the planet, and not affected by the angle of projection. This is because the angle of projection only affects the object's initial velocity, but it does not change the total energy of the object.

In summary, the escape velocity is independent of the angle of projection because it is solely determined by the mass and distance of the celestial body, and the total energy of the object remains constant throughout its trajectory.