Where does the energy go?

What do you mean by this? Remember that acceleration is a vector quantity, and also that v=0 does not imply a=0, and so in your case you have a change in acceleration.

 

When you make something to go at a high speed, you give it some kinetic energy. When you try to stop it, the energy normally turns into heat, another type of energy.

 

You did work on the object when accelerating it. The object did work on you when decelerating it.

Think of it this way: compress a spring, doing work to store potential energy in the spring. Now use that spring to accelerate your object.

Quickly run around in front of the object, place your spring in front of it and use the spring to slow it. Ignoring resistance or any kind of friction, the kinetic energy in the object will be exactly enough to compress the spring back to where it was before. If you don't like the idea of "running around in front of the object", imagine two different but identical springs. Same thing happens. Net result, you did work to compress a spring initially, and now you are back in that same situation. No energy change.

 

The answer here is that the kinetic energy you put into the object was later transformed either into potential energy somewhere else, or into heat. For example, if you use a spring to stop the object, the kinetic energy from the formerly moving object will be transformed into spring kinetic energy. Or if you stop an object by friction (or some other non-conservative force), then the kinetic energy will be lost to friction. I could give slightly different answers based on the mechanism used to stop the object, but this is the basic idea.

 

Yes and no. They are not the same thing. However, you may notice that momentum "p" is the derivative of kinetic energy with respect to velocity:

[tex]\[
\begin{array}{l}
E_k = {\textstyle{1 \over 2}}mv^2 \\
p = mv \\
\end{array}
\]
[/tex]