In the case of free fall, we can take our x axis to be in the
horizontal direction and the y axis to be in the vertically
up. The acceleration of gravity is then 
Instead of writing 9.81, let's just call it g, so 
.
We can write down eqs. 5.12 and 5.13
in component form. Again we'll follow the standard notation of
using subscripts x and y to denote the components of a vector
quantity, so
Note that the x component of the velocity is constant, which it should be since there's no acceleration in that direction.
Now for position:
and
This says that motion of an object in free fall has its x and y components
completely decoupled. That is they're independent of each other.
If you change 
it doesn't affect the y motion at all.
The motion in the x direction is that of an object going at a constant
velocity. The motion in the y direction is exactly the same equation
as in one dimension for a particle in free fall. By combining these
two components, you get the overall motion.
