How does mechanical energy conservation work? I am a little confused about how mechanical energy conservation operates when it comes to things like predicting velocity. I know that if conservative forces are the only forces acting on a body, then we can say that mechanical energy is conserved. This is simple to see when we have lateral up and down motion, but when it comes to predicting the velocity of a pendulum or a roller coaster (neglecting all friction) I'm not sure how the law operates. For example, given the initial peak height of the roller coaster, I can predict the velocity at any point, despite the fact that there are various loops and curves. And for a pendulum, the motion is in an arc. Despite these complexities, the same equations used for these situations are used for simple free-falling situations. Could someone give me a deeper understanding of how these equations are able to make predictions about velocity and such in complex situations like riding a roller coaster?
 A: One thing that might help you address your question is that conservation of energy is deeply connected with the idea of path independence. If you have a force field--say, gravity--it can be "conservative" (conserves energy) or "non-conservative" (...guess). An example of a force that is not conservative is friction, because a body experiencing friction loses energy.
Now, I'm guessing from your question/age in the profile that you are in an early physics class, and may not know calculus. But in calculus, we can take a sum over a path that a body goes on--imagine that at every point in space we ask ourselves "how much work did the force do on our moving body" and then we add those all up. A conservative force has the remarkable property that whenever we do this, the path doesn't matter--only the beginning and end points of the path matter. It's like if you're moving around a city, take a crazy route to get to where you're going, and ask how far you are from the beginning. It doesn't matter how you got there, to determine the distance now only requires knowing the start and end points.
Because gravity (and electromagnetism, and others) is conservative, it doesn't matter how complicated the roller coaster is. It doesn't matter what weird path you go on. The energy equation only cares about $mgh$, and the change in energy only cares about $mg (h_2 - h_1)$. That's it. If you did a weird squiggle along the way, your final state is actually totally unaware of that. All it knows is how far you've come.
A: 
For example, given the initial peak height of the roller coaster, I
  can predict the velocity at any point, despite the fact that there are
  various loops and curves

It is very simple: if you know the height $h$ of a body you know its potential energy which is $mgh$. This energy is given to the body (transformed into Kinetic energy) at every lower point, and when it reaches the ground its velocity will be $v_G=\sqrt(2g*h)$. Supposing the peak is at $20m$ its velocity will be $20m/s$
But you can also find the velocity at any other point, calculating the energy the body has acquired considering the difference in height $v_{h_1} = \sqrt(2g* (h-h_1)$
A: Let's look at the example of roller coaster. At a given moment, 2 forces are acting on the train: 


*

*Gravity: towards $-z$ direction.

*Normal force: direction normal to the rail.


The normal force does no work, since work is given by $dW=\vec{F}\cdot d\vec{x}$, and the normal force is always normal to the direction of the rail, thus normal to the moving direction of the train. Therefore, the work done to the train is solely from gravity. and amount of work it has done is given as
$dW=\vec{F}\cdot d\vec{x} = -mg\, dz$
and it is path independent.
In general, one can classify forces acting on a body into two categories.


*

*Force tangential to the velocity

*Force normal to the velocity.


Every force that is tangential to the velocity always changes the kinetic energy of the body, while every normal force changes the moving direction of the body. 
In the examples you provided, there seem to be some other forces acting on the body, but in their nature, they all, except for the gravity, are normal to the moving direction of the body. What they do is changing the direction of the movements. 
So, if your concern is just the kinetic energy, what matters here is gravity only, since it is the only force that can be applied to the tangent of the moving direction. And as mentioned above, its amount of work is independent of the moving direction.
A: mechanical energy conservation is sum of all  conservatives energy such that there is no external forces like frictional or an explosion.
basically we mainly deal with 

Blockquote

GRAVITATIONAL energy and KINETIC energy

Blockquote

.
mgh + 1/2m sq.(v) = constant 
when no other external forces like i perviously mentioned do not act. if they do so then add them also
when it comes for solving question related to roller coaster take frictional force acting btw the tires like we do in rotation.
A: When someone moves in a field then the work generated by field forces is independent of motion and related to the starting and ending positions. The quantity $U$ 
$$\int_{r_1}^{r_2} \mathbf F( \mathbf r) \cdot d \mathbf r =   [U ( \mathbf r)]^{r_2}_{r_1}$$ is called potential energy and is associated with a potential $V$.By work energy thm we get $$dK = W = W_0 - dU$$ and when $W_0$ is 0 then the quantity $E=K+U$ is conserved. 
