Modeling projectile motion as subject to a constant accelera

[Anama Skout]

Consider the following diagram of a projectile motion ($\hat{\bf i}$ and $\hat{\bf j}$ are the unit vectors of the $x$ and $y$-axis respectively)

We know that $$F=ma.\tag1$$ This can be rearranged to $$a=\frac Fm.\tag2$$ So there are actually two accelerations, one with magnitude $mg$ and another with magnitude $\frac Fm$. However when we model that projectile motion using the equation $$x_f=x_i+v_it+\frac12at^2,\tag3$$ we put $a=mg$ and not $a=mg+\frac Fm$. Why is this the case? What am I missing?

 

[nasu]

What is this F?
After the projectile leaves the cannon there is no force on it besides gravity.

 

[Anama Skout]

What is this F?
After the projectile leaves the cannon there is no force on it besides gravity.

$F$ is the force that the projectile exerted on the ball.

 

[nasu]

So you have a projectile and a ball? Isn't the ball the projectile, the moving object?

 

[Anama Skout]

So you have a projectile and a ball? Isn't the ball the projectile, the moving object?

Hmm sorry (english isn't my mother language) I meant by the projectile the canon, the thing that projects, and the ball the thing that was projected, the moving object.

 

[nasu]

Oh, OK.:)
As long as the ball is inside the cannon there is indeed a horizontal force on it. Once the ball is out of the cannon, this force does not act anymore.
When we study projectile motion, we consider as initial state (with initial velocity) some position outside the cannon, where there is no more force exerted by cannon on the ball.