The previous post was quite strong. The essence of the scene is that Brian is desperately trying to land the plane. He has received a small amount of instruction from the pilot, who is now dead. The jerking of the plane is what results. However, Brian has enough skill or knowledge to ensure that the plane of the nose is tilted down in order to facilitate a landing in a clearing of the forest and not against the trees. His throwing up is a result of two things. The violent and intense landing of the plane is one reason, as it caused an extreme amount of discomfort as altitude dropped. The second reason might be the height of anxiety Brian experienced at the notion of trying to navigate a safe landing, being only suddenly thrust into the situation and with a dead pilot next to him.
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).
In the book "Hatchet" Brian is flying over the Canadian wilderness in a single engine Cessna. The only people in the plane are Brian and the pilot. The pilot has shown him some things about the plane after he sees that Brian is somewhat teary eyed. Brian's eyes were misting up when he was thinking about his parents divorce. During the flight the pilot has a heart attack. Brian has to fly the plane because the pilot is unconscious possibly dead at the time. The plane begins to dive and that is when he uses whatever skill and little knowledge he has just learned to get the plane to stabilize so it won't crash. Brian keeps the plane in the air for several hours but the gas is running out. He has very little gas left and has to make a final decision to land the plane or die or possibly die while landing the plane. His stomach knots up and he throws up.
"tighten into a series of rolling knots"
I believe he throws up from a combination of the plane movement and the stress of the situation.