# Pushing force on wagon

Let's say that you are pushing a wagon by applying a force $\vec{F}$ from behind, as such:

Now, according to Newton's second law, the center of mass of the wagon should accelerate, because the sum of the external forces is not $\vec{0}\text{ N}$. If we assume that the coefficient of friction between the wheels and the ground is $0$, then I fully understand what's going on - $\vec{F}$ is the only applied force (or the sum of all the forces) on the particle system, which causes the CM to accelerate.

However, the problem arises when I take into account the cases where the coefficient of friction isn't $0$. In these cases, the wheels will presumably begin to rotate. Since this rotation is due to friction, it seems likely that there is a frictional force (let's call it $\vec{f}$) on the wheels from the ground. However, if such a force is present, and the initial force remains unchanged, then according to Newton's second law, the acceleration of the wagon should be given by:

$$\vec{a} = \frac{1}{m_{\text{wagon}}}(\vec{F} + \vec{f})$$

So if this is the case, wouldn't that make the acceleration different from the case where the coefficient of friction between the surfaces is $0$? In picture form, this is pretty much what I mean is the case generating the second equation:

Moreover, I don't even know whether the direction of the frictional forces in the above picture is correct or not (i.e. whether $k_1, k_2 > 0$), which kind of goes to show that I'm fairly confused about this whole thing. Intuitively, it feels like $k_1 < 0 \wedge k_2 < 0$, since the ground "holds the wheels back", but in that case, won't friction "slow the wagon down" rather than helping it roll?

I'll summarize my questions:

1. When $\mu \neq 0$, which forces are present on the object?
2. If there is a frictional force, which direction does it have?
3. How does this force affect the acceleration of the wagon's CM?

Moreover, I don't even know whether the direction of the frictional forces in the above picture is correct or not (i.e. whether $k_1, k_2 > > 0$), which kind of goes to show that I'm fairly confused about this whole thing.