The acceleration of the centre of mass 
Given two balls that are thrown straight up in the air at a speed of $40 \:\mathrm{m/s}$, $0.10 \:\mathrm{s}$ apart. One ball is twice the mass of the other. The heavier ball is thrown second. 

I'm asked to find the acceleration of the centre of mass at the following times:


*

*When the two balls are moving up 

*$0.10 \:\mathrm{s}$ after the lighter ball lands
For the acceleration of the centre of mass I'm using this formula:

I'm assuming that the acceleration of the two balls will be equal to $9.8\:\mathrm{m/s^2}$
Plugging all the values for first case:


*

*$a_{cm} = \frac{m*9.8+2m*9.8}{m+2m}$ which will give me $9.8\:\mathrm{m/s^2}$


For the second case


*The acceleration of the lighter ball will be equal to zero, as it will be on the ground


Hence, plugging all the values in the formula:
$a_{cm} = \frac{0+2m*9.8}{2m}$ which will also give me $9.8\:\mathrm{m/s^2}$
However, it does not make sense for me that the acceleration of the centre of mass will be equal in both cases. 
Am I assuming that their acceleration is $9.8\:\mathrm{m/s^2}$ mistakenly? 
 A: There is a mistake in equation (2). Its denominator should include the total mass of the system that you're considering, so the denominator should be '2m+m'. You correctly used this value for equation (1), but apparently incorrectly believed that since the position (and velocity?) of the lighter mass 'm' is zero that the value of 'm' shouldn't be included in the denominator of equation (2). 
As a general rule remember that coordinate systems do not have a physical reality in and of themselves but are merely an artificial convenience that we (human beings) use to determine and calculate physical quantities. Real physical quantities are independent of coordinate systems. If you find that the value you are getting for some physical quantity depends on the particular coordinate system that you are using, then that's a clue that there is a mistake somewhere.
A: In the second case, you made a mistake in the denominator. You always put the overall mass. The lighter mass is part of the system that you are trying to find the center of mass of, even when it has a zero value of acceleration. So, the denominator should be 3m.
In general, we put in the denominator the mass of every body that is in the system of bodies for which we want to find the center of its mass.
