Suppose that I have a box in space and it is filled with air in it. I keep a fan at one end and blew the air to other end. The air will hit the box with some velocity, so it should move.
I know that this is completely false because the momentum of the system remains constant. But somehow I cannot explain that clearly.
If you use Newton's 3rd law of motion, which states that for every action, there is a equal and opposite reaction i.e. when the fan accelerate the air particles by a force, it also experiences a backward reaction force and according to 3rd law of motion these action and reaction force must be equal.Hence the net force acting of the box is zero . That's why it is not moving by the fan.
$F_{reaction} = F_{action}$
If you want to explain it by the momentum conservation theorem then you have to consider the change of the fan and air particles and the summation of change of momentum of all particles must be zero. Since here the velocity of the fan and the air particles are opposite directed, hence the summation becomes zero.
$\sum^n_{k=1}$ $ m_i\bar{v}_i$
No! You have to make an opening in the front and in the back. The theory is that the sum of the momentum of all moving particles and things $\displaystyle\sum_{k=1}^nm_i\mathbf{v}_i$ should be zero.
If some particles are accelerated by the fan and moving backwards a force will be applied to the box in direction forward.
If it wasn't a smooth stream of moving molecules, but for instance rather heavy balls, the box would be shaking forward and backward when the balls was thrown and when they hitting against the sealed box.
