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.
 A: The best way to prove something is wrong, is by performing a simple experiment, giving a counterexample. Take two identical objects (balls, pens, books). Throw one of the objects upwards and the other object downwards, so they have different initial velocities. The moment you let them go, they are in free fall. I am quite convinced the latter one will be one the ground first.
A: One thing to remember is that horizontal velocity has nothing to do with the time to hit the ground. If one ball falls straight down, and the other is thrown horizontally (with zero vertical component), they will still hit the ground simultaneously. The thrown ball will simply fall further from the building. I suspect that's what the teacher meant with the "different velocities".
Of course, this is only an approximation. As the earth is not flat, the ball thrown horizontally will hit the ground a tiny bit later than the vertical one. If the horizontal velocity is 7 km/s or more then the ball will not hit the ground at all. It will fall around the earth, which is another way of saying that it will be in orbit. And, at more than 11 km/s the ball will leave the earth forever. And even this is not quite right: unless there is no air friction, at 7 km/s ball would burn up...
A: 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!
