Free falling objects

My teacher and I are in the middle of an argument because she says that if you were to drop two objects at the same time and the same height, but with different initial velocities, both of them would hit the ground at the same time. She also said this was proven by Galileo's unregistered experiment when he let two objects fall from the Leaning Tower of Pisa. So I told her that she was right, they would fall at the same time only if both initial velocities had been the same, and that Galileo's experiment was mainly to prove that mass has nothing to do with the timing registered on two objects falling from the same height.

Could you please help me out, using 'pure' physics to prove my teacher wrong? I've already searched the internet, but I haven't found something that would convince her because it's what I already have explained to her, but she's in denial and I'm thinking she just doesn't want to accept that she's wrong.

Assume no drag/air resistance.

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Buy a trip to Pisa with your teacher. Stand together with her beneath the tower and ask someone at the top to simultaneously throw a rock on you, and shoot at her with a gun. She will be shot but before you will be hit by the rock, you will go away. You will survive and you will get rid of your wrong teacher, too. –  Luboš Motl Oct 22 '12 at 18:20
@LubošMotl If this were a physics teacher, this is by far the best approach. But, let them throw a bullet, instead of a rock, just to have a fair comparison :) –  Bernhard Oct 22 '12 at 18:27
Ask her if she have ever heard about circular or escape velocity and satellites which never return back to the ground (neglecting air resistance, as you both did) –  Leos Ondra Oct 22 '12 at 19:18
Silly schools letting history majors teach physics. When will they learn. –  ja72 Oct 23 '12 at 14:13

If we are throwing two objects directly to the ground you are right. So from our kinematic equations:

$$V_f = V_i + at$$

I would ask your teacher. What happens to the $V_f$ if $V_i=0$? Then Follow it up with what would $V_f$ be if $V_i$ was very large?

The initial velocity DOES have an effect here.

HOWEVER: Make sure that you are not misinterpreting the question! If you are throwing two objects in something that is sort of parabolic arc (i.e. a projectile) and you threw them at the same time, the time it would take for each one to hit the ground would be THE SAME.

This is because at their peak the objects both only have one thing accelerating them (gravity) and because at their peak their V = 0! This is very un-intuitive and you should look into exploring it!

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So if I threw two balls horizontally from a building at the same time, one with V and the second one with V/2, they would fall at the same time? –  Jerson Zuleta Oct 22 '12 at 21:10
@JersonZuleta Yup! Find a safe spot and perform a similar experiment yourself. Launch one object as close to horizontally as you can, and simply drop another object from the same height at the same time. –  kleingordon Oct 23 '12 at 9:07