escape speed

Notice that we are considering motion on a trajectory up to infinity. At very large distance from earth, the three paths shown are more or less the same, they are going away from the centre of the earth.

It is only at short distance from earth that the three paths show differences. Path 1 and 2 would become curved because of the direction of earth attraction. As distance increases, the angle between the motion direction and the gravitational force decreases, and eventually the paths of 1 and 2 would be similar to that of path 3.


2010-06-27 17:09:44 補充：
You could just imagine that if a spacecraft is sent to outside the solar system from earth. When the spacecraft is far away from the solar system, the size (diameter) of the earth is negligible compared with the distance of the spacecraft from earth....

2010-06-27 17:11:32 補充：
...As such, the direction of the trajectory on firing from earth is unimportant. Launching paths 1, 2 or 3 will then end up at the same final trajectory.

還是真正用數學推導最為實際，因為可能只是「大約地無關」而不是「真正地無關」。

physics textbook當然隻字不提，因為在physics textbook提及de絕對是對沒有de底子的學生不公平。

We know from a previous question regarding Kepler's first law that an object influenced by a central gravitational force follows one the trajectory below.

圖片參考：http://imgcld.yimg.com/8/n/HA05726829/o/701006270223113873433090.jpg

where the eccentricity was calculated to be

圖片參考：http://imgcld.yimg.com/8/n/HA05726829/o/701006270223113873433101.jpg

Here L = angular momentum, and the energy is defined by the sum of KE and PE

圖片參考：http://imgcld.yimg.com/8/n/HA05726829/o/701006270223113873433102.jpg


Hence, whenever the energy is non-negative, the object follows an open trajectory (parabolic for E=0 or hyperbolic for E positive). It will therefore escape and never return, regardless of the direction of initial velocity.

因為物體嘅gf永遠指向地心，同方向無關，如果速度未達逃逸點，你只能圍繞地球轉，當你能超越逃逸速度，eg.11km/s，你將可以擺脫gf以切綫(tangent)飛出引力圈。

Yes, first of all, energy is scalar which is independent of direction.
Refering to your example,
For the 1and 2 direction:
Assume the object runs in a circular path first. But the centripetal force is not enough to keep the circular trajectory on account of the speed. The object will run in a circle with greater and greater radius. At the same time, the speed of object will get lower (no matter the horizontal and vertical speed). Therefore, it's kinetic energy is changed into potential energy, which is same as the deduction in the above formula.

Therefore, they have the same escape speed