newtonian mechanics - What does an applied force on an already accelerating object do?

I'm a big beginner having only taken Grade 11 high-school physics.

Imagine this situation: an object is accelerating north, and while it's accelerating, it's 'hit' by a northward force.

To me it would make sense that the acceleration of the object would increase - something like this:

aNew = aOld + (f/m)

I got the (f/m) from f=ma.

But then think of gravity - while an object is accelerating downward at 9.8m/s2, it has the force of gravity pulling it too. So then, based on my previous logic, it's acceleration would constantly be increasing!

aNew = aOld + (fGravity/mass)

And I know that you don't increase your acceleration while you fall, so now I'm confused.

In a nutshell: What exactly are the rules for how an already accelerating object reacts to forces? Am I wrong that your acceleration would increase?

EDIT: I realized I've made a big mistake; I never knew that the acceleration drops to 0 the second the net force is 0. This makes sense now.

Answer

The acceleration of an object is a result of the sum of all the forces. If it was accelerating before you hit it, there must have been a force. If an object is falling in air, there are forces of gravity and air friction on it. When all forces even out the object continues with the same velocity.

Simply take the vector sum of all forces - that will tell you what is going on.