How Do Final Speeds of Two Balls Compare When Thrown from a Building?

[aquapod17]

Homework Statement

A student at the top of a building of height h throws ball 1 upward with a speed of v_i and then throws ball 2 downward with the same initial speed, v_i. How does the final speed of ball 1, v_1f, compare to the final speed of ball 2, v_2f, when they reach the ground?
v_1f = ______ multiplied by v_2f

Homework Equations

a = delta v/delta t
v_f = v_i + a(delta t)

The Attempt at a Solution

I drew a picture showing the person throwing ball 1 upward and then it descending and ball two falling to the ground.

 

[Rake-MC]

Okay well let's think about this. We know if there is presence of uniform force in only one dimension then the trajectory is going to be parabolic. For ball 1, it will leave his hand at v_i go up in its arc, reach a maximum height (where vertical velocity is zero) begin to fall again with identical acceleration as before.
This implies that when it passes the thrower for, it will have the same velocity it was thrown with, but in the opposite direction.

 

[baba89]

Then I wrote out the equations for velocity and acceleration. I also noted that the acceleration due to gravity is constant and equal for both balls.

Based on the equations, we can see that the final velocity of ball 2 will be equal to the initial velocity of ball 1. This is because both balls start with the same initial velocity and experience the same acceleration due to gravity. However, the final velocity of ball 1 will be negative, as it is moving in the opposite direction of its initial velocity. Therefore, the final speed of ball 1, v1f, will be equal to the negative of the final speed of ball 2, v2f. In other words, v1f = -v2f. This means that the final speed of ball 1 will be equal in magnitude but opposite in direction to the final speed of ball 2.