Why Is Initial Velocity Equal to Final Velocity in Parabolic Motion?

[lilstar]

Homework Statement

I was wondering, when an object's motion is in the shape of a parabola, and it lands at the same height that it was shot from.. why would the initial velocity and the angle from the ground at which it was shot from.. be equal to the final velocity at which it hit the ground?

Homework Equations

The Attempt at a Solution

Would it be because of its parabolic shape?

 

[kbaumen]

Because both when moving upwards and downwards the same gravitational force is working on the object. It's acceleration is always ~9.8 m/s$$^{2}$$ and directed to Earth's center.