physics behind roller coaster

Gravity is an important component of a roller coaster, the gravity pulls the roller coaster down to the ground, so that so roller coaster won't move vertically forward.
For a non-idealized roller coaster system, not all of the energy is conserved.  Friction is the main cause of energy leaks in the system and the reason why mechanical energy is not fully conserved for a real roller coaster.  This is because friction is a nonconservative force.  Nonconservative forces are forces that cause a change in total mechanical energy.  Friction opposes motion by working in the opposite direction.  The friction between the train and its tracks as well as between the train and the air take energy out of the system, slowing the train and creating both heat and sound.  This effect is most noticeable at the end of the ride as all remaining kinetic energy is taken out of the system though brakes.  Because of the energy leaks due to friction, each successive hill or loop on a roller coaster must be shorter than all the hills or loops previous to it, otherwise the train will not have enough energy to make it all the way over.

vector addition

     When we are doing vector additon question, we basically just add the X and Y. For each vector, we calculate the X and Y by doing the math, we use sine or cosine to calculate the value for "x" and the value for "y". And we add all the X together, also, add all the Y together, to obtain the final result Dx and Dy. Once we get that value, we can simply get the final result by using the pythagorean theory, and then use tangent law to calculate the angle.
And That is the Final Answer /!!!