Tangential and radial acceleration in projectile motion I'm currently learing kinematics, specifically projectile motion and as an example in my textbook is a bullet fired at some angle. I understand the derivation of formulas to describe that motion and that the only acceleration acting on the bullet is $g$ directed downwards. But then it states that $g$ could be "broken" into tangentail and radial component. There is no picture how that would look, but wouldn't that mean that $g$ is not the only acceleartion acting on the bullet, but instead there is its $x$ component and $y$ component? And if that is so why then $x$ component isn't being used when calculating the speed in the $x$ direction?
 A: @ Nick Basically you are confusing two different coordinate systems. In the coordinate system where you decomposed g into two components, there the question of x and y doesn't seem valid because you have transformed the cartesian coordinate system into a different one. You would have new x and new y there and you have to stick with one while solving the kinematics. 
You are free to choose any of the coordinate system however analysis in the reference frame (more technical term) where you decomposed the gravity into two components is a bit more typical the reference frame itself is changing orientation and you would have to take that into account if you are inclined to solve in that frame.
A good and similar example of this would be to consider motion on an inclined plane. When a body is undergoing motion "along" the inclined plane, it is far more convenient to solve in a reference plane which has axis (x and y) parallel and perpendicular to the inclined plane. And thus since you fixed your reference frame now, you have to resolve all the forces etc. This is purely a simplification or a transformation from the normal cartesian system to the inclined plane one. And as mentioned above, you can also have polar coordinates. 
Before solving any kinematics or dynamics problem, fix the coordinate system (and this fixes your convention too - positive and negative) and then start resolving forces, velocities etc. A natural question arises then how to choose which frame to solve in and avoid this confusion of mixing up two or more frames? The answer is practice and experience. You would yourself start seeing that its tedious and highly complex to solve in one frame and thus would choose to go with the simplest solving. That's Physics, not mathematics ;)
A: Think about a picture of someone throwing a stone at some angle $\varphi$. The acceleration due to gravity is downward, as you said. 
Now, we're in a 2D plane, so we can decompose this vector in two components. There are different ways of doing this. One can use a Cartesian coordinate system, i.e. use some $x$- and $y$-axis and decompose the vector in projections onto those axes. The projections are then the $x$- and $y$-components of the vector. 
Alternatively, we can use polar coordinates. We should then define the origin of our coordinate system, which will allow us to decompose the acceleration vector into a radial and a tangential component ($r$- and $\theta$-component) of the acceleration. Note that, for this coordinate system, the decomposition depends on where you define your origin. This is not the case for a Cartesian coordinate system. You could draw a picture of this (choosing, for instance, the origin at the position from where the stone is thrown), this might make it more clear why the decomposition depends on the location of the origin in polar, but not in Cartesian, coordinates.
