Troubles with a Dynamics exercise

[Felafel]

Homework Statement

Two carts (1&2) on a flat surface, are pushed by an external force ($\vec{F}$), exerted on 1 (the carts are motionless and touching each other).

Consider the two objects as particles and take no notice of any friction.

F=12N; mass of 1 ($m_1$)=4,0 kg; mass of 2 ($m_2$)= 2,0 kg.
Find the intensity and the direction of the force exerted by 1 on 2 ($\vec{F_{12}}$) and the force exerted by 2 on 1 ($\vec{F_{21}}$)

The Attempt at a Solution

I tried solving the system given by:
$\vec{F_{12}}=\vec{F} - \vec{F_{21}}$ and $\vec{F_{21}}= m_2 * a$

obtaining:
$m_1 * a = 12 - m_2 *a$ $\Rightarrow$ $a=2,0 m/s^2$

and thus:
$\vec{F_{21}}=2,0kg * (-2,0 m/s^2)=-4 N$ with the minus sign, as this force is opposite to F
$\Rightarrow$ $\vec{F_{12}}=16N$
which, according to my textbook is not the right result.
I don't get where are the mistakes, though. Can anyone help me please?

 

[ap123]

Your first equations are not right - you're mixing forces acting on different objects.
You need to draw some free-body diagrams and then apply Newton 2 to them.

What will be the relation between the 2 forces you're asked to find?

 

[Doc Al]

$\vec{F_{12}}=\vec{F} - \vec{F_{21}}$

This violates Newton's 3rd law.

 

[Felafel]

allright, should it be like this then?
i find the acceleration, which is:
$a=\frac{\vec{F}}{m_1+m_2}$ = $2 m/s^2$
in the free-body diagram of 1 there is $\vec{F_{21}}$
so I multiply the acceleration for $m_2$, which gives $\vec{F_{21}}=-4N$ (because its direction is opposite to that of the x-axis)
and, for Newton 3, there must be an equal and opposite force, which means $\vec{F_{12}}=4N$

 

[ap123]

allright, should it be like this then?
i find the acceleration, which is:
$a=\frac{\vec{F}}{m_1+m_2}$ = $2 m/s^2$
in the free-body diagram of 1 there is $\vec{F_{21}}$
so I multiply the acceleration for $m_2$, which gives $\vec{F_{21}}=-4N$ (because its direction is opposite to that of the x-axis)
and, for Newton 3, there must be an equal and opposite force, which means $\vec{F_{12}}=4N$

Yes, this looks right :)
You can check the answer by seeing that the resultant force on m₁ is therefore 8N which agrees with its acceleration.