Baseball Throws: Energy Conservation & Solutions

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The discussion revolves around the application of energy conservation principles to baseball throws, specifically regarding final velocities. The user attempts to prove that all final velocities are equal by using energy conservation equations, leading to the conclusion that the final velocity can be expressed as v_f = √(2gh + v_i²). They assert that since initial velocities, gravity, and height are constant, all final velocities should also be equal. Despite this reasoning, they encounter issues with an automated submission system and express frustration with the homework platform. The consensus is that their reasoning is correct, but technical issues are hindering their submission.
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Homework Statement


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Homework Equations


Energy conservation et al.

The Attempt at a Solution


I think that all of the final velocities will be equal, but I am not sure how to show this mathematically. Seems like a trick question.
 
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mgh_{i} + \frac{1}{2}mv_{0}^{2} = mgh_{f} + \frac{1}{2}mv_{f}^{2}
gh_{i} + \frac{1}{2}v_{0}^{2} = gh_{f} + \frac{1}{2}v_{f}^{2}
Final potential energy is zero at the ground, so:
gh_{i} + \frac{1}{2}v_{0}^{2} = \frac{1}{2}v_{f}^{2}
Which gives that:
gh_{i} = \frac{1}{2}v_{f}^{2} - \frac{1}{2}v_{0}^{2}
 
I see that it turns into a familiar kinematic equation. I end up with:

v_{f}= \sqrt{2gh+v_{i}^{2}}

Since the initial velocity of all the balls is the same, gravity is the same, and the displacement, well height, is the same, wouldn't that give me that the final velocity is equal to a constant in all cases, and thus they are all equal? Is this correct thinking?
 
I put them as all the same, and it's incorrect? What mistake am I making here?
 
Your thinking is correct--they are all the same. Your automated answer submission machine is having a fit.
 
It figures. I sent my professor an email on the question. Masteringphysics is so annoying. Thanks for the help.
 
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