Force and Accleration It's just a basic question I had when I was studying physics years back,
So acceleration have two equations
$$a=\frac{F}{m}$$
and
$$a=\frac{\text{d}v}{\text{d}t}$$
So by the first equation, if I'm pushing a wall, the wall does have an acceleration.
But by following the second equation, since velocity is proportional to distance and also because the wall has not moved an inch, the acceleration is zero.
What does this mean actually?
 A: Both are perfectly compatible and coherent indeed. If the wall is fixed on the floor your force on the wall, $F$, is counteracted by an equal and opposite force coming from the fixing ground to the wall $-F$. This is Newton's third law. It's important to note that in $F = ma$ (Newton's second law) the force is really the sum of all forces applied to the object in question and the accelleration is really the sum of the accellerations imparted by each force. You can have a huge force as an imput but if the interaction is set in such a way that the reacting force acts on the same object (by fixing the wall to the floor) you end up with both accellerations canceling out just like the two forces (your input force and the response of the floor on the wall) cancel out.
What about a scenario where the wall is not anchored to the floor? Well, it is not true at all that if you have some force $F$, you will notice some accelleration $a = F/m$. That's because the mass of the object $m$ could be gigantic. If the mass is huge your force will produce a tiny accelleration, to the point it might even be undetectable. You need to account for the inertia of the object (its resistance to motion) and not only the force if you want to fully understand what the accelleration would look like (because $a = F/m$ depends both on the $F$ and the $m$).
A: To add to the other correct answers here, I think it's also useful to point out that
$$\mathbf{a} = \frac{\mathbf{F}}{m}$$
is a dynamical equation. It describes what the acceleration on a body will experience when there is an external force $\mathbf{F}$. However, the equation $$\mathbf{a} = \frac{\text{d}\mathbf{v}}{\text{d}t}$$
is a kinematic equation. It describes how the body will change its instantaneous velocity due to this imposed acceleration.
The  kinematic equation is always true, it is the very definition of acceleration. The dynamical equation is true because of Newton's Second Law. It is not true in situations where this law does not hold.
In this case, however, Newton's Second Law does indeed hold, but as the other answers have pointed out, the net force on the wall is zero, and hence it has no acceleration and consequently its velocity does not change.
