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.

 

[ogb p]

I would approach this question by first analyzing the concept of projectile motion. When an object is launched with an initial velocity and an angle from the ground, it follows a parabolic path due to the force of gravity. This path is symmetrical, meaning that the object reaches its maximum height at the midpoint of its trajectory and then falls back to the ground at the same height it was launched from.

Now, let's consider the velocity of the object at different points along its trajectory. At the initial point of launch, the object has a certain velocity in the horizontal direction and a certain velocity in the vertical direction. These two velocities combine to give the resultant velocity of the object.

As the object travels along its parabolic path, the vertical component of its velocity decreases due to the force of gravity pulling it down. However, the horizontal component of its velocity remains constant. This means that at any point along the trajectory, the resultant velocity of the object will always have the same horizontal component as the initial velocity, but a decreasing vertical component.

When the object reaches the same height it was launched from, it has the same vertical component of velocity as it did at the initial point of launch. However, the horizontal component of its velocity remains unchanged. This means that the resultant velocity at this point will be the same as the initial velocity, with the same angle from the ground.

In summary, the reason why the initial velocity and angle from the ground are equal to the final velocity at which the object hits the ground is because of the symmetrical nature of projectile motion. The horizontal component of velocity remains constant, while the vertical component decreases and then increases again, resulting in the same resultant velocity at the beginning and end of the trajectory. This is true for any object launched at the same angle from the ground and with the same initial velocity, regardless of its shape.