Today, in my high school physics class, we had an introductory class on electromagnetism. My teacher explained at some point that an object with a very high speed (he said it started to get somewhat clearly noticable when travelling at 10% of the speed of light) will gain mass, and that that's the reason why you can't go faster than light.
One of my classmates then asked, why is this so? Why does an object with higher speed gain more mass? This of course is a logical question, since it is not very intuitive that a higher speed leads to a higher mass. My teacher (to my surprise (responded saying that it is a meaningless question, we don't know why, in the same way we don't know why the universe was created and those kind of philosophical questions.
I, being interested in physics, couldn't believe this, I was sure that what he said wasn't true. So after a while of thinking I responded saying:
Can't we describe it with Einstein's $E=mc^2?$ If an object gains speed, he gains more (kinetic) energy. With this equality we see that the more energy an object gets, to more massive it becomes.
He then replied saying that this formula is used for different cases, whereupon he gave a vague explanation as to when it is used. He gave me an example to show what I said was incorrect; when a car goes from $10 m/s$ to $40m/s$, according to what I said we would see a big increase in mass, and we don't (this sounded logical to me). So here I am, with the following questions:
Why does an object with a higher speed have more mass (than the same object with a smaller speed)?
When is $E=mc^2$ used and why is my argument incorrect in explaining this phenomenon?