Baseball Throws: Energy Conservation & Solutions

[QuarkCharmer]

Homework Statement

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.

 

[QuarkCharmer]

$$mgh_{i} + \frac{1}{2}mv_{0}^{2} = mgh_{f} + \frac{1}{2}mv_{f}^{2}$$
$$gh_{i} + \frac{1}{2}v_{0}^{2} = gh_{f} + \frac{1}{2}v_{f}^{2}$$
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:
$$gh_{i} = \frac{1}{2}v_{f}^{2} - \frac{1}{2}v_{0}^{2}$$

 

[QuarkCharmer]

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?

 

[QuarkCharmer]

I put them as all the same, and it's incorrect? What mistake am I making here?

 

[lewando]

Your thinking is correct--they are all the same. Your automated answer submission machine is having a fit.

 

[QuarkCharmer]

It figures. I sent my professor an email on the question. Masteringphysics is so annoying. Thanks for the help.