In the book Hatchet, why does Brian push the nose of the plane down and throw up?

In the book Hatchet, Brian pushes the nose of the plane down because the engine has suddenly died, and that is the only way to keep it flying a little longer. He throws up most likely because of terror - the terror of knowing that he is going to crash-land, and possibly die.

Brian has no idea of how to fly a plane, but he has had quite awhile to think about it after the pilot dies. The plane has been flying smoothly on its own during the period since then, and in that time, after his initial panic, Brian has tried to prepare himself for what he knows is coming. He figures that eventually, the plane will run out of fuel and the engine will stop, and he guesses

"that without the propellor pulling he would have to push the nose down to keep the plane flying - he (thinks) he may have read that somewhere, or it just (comes) to him. Either way, it (makes) sense."

Brian's conjecture that he will have to push the nose down to keep flying speed when the engine stops, then pull it back up to slow the plane as much as possible before impact is correct. When the engine finally dies, he does just as he has planned, but as he pushes the nose of the plane down, the terror and the finality of what is about to happen causes his stomach to "tighten into a series of rolling knots" and he throws up (Chapters 2 and 3).