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why would negative initial velocity equal final velocity of the ball at the maximum range of the ball?

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Answer:

Negative initial velocity doesn't equal the final velocity of the ball at the maximum range of the ball.

The final velocity of the ball at the maximum range is equal to zero, since at that point the ball has reached its highest point and is no longer moving upward, it is now moving downward.

In physics, when an object is launched upward, its initial velocity is positive and it slows down as it reaches the highest point, then its velocity becomes negative as it starts to fall. The final velocity of the object is zero at the highest point as it is no longer moving upward and it's not yet moving downward.

In the case of a ball being thrown, it's initial velocity is positive when it's thrown, and it slows down as it reaches the highest point, then it's final velocity is zero at the highest point and negative when it starts to fall.

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