The force doesn’t have to push against anything. For example, gravity pulls us towards the earth without pushing against anything else. Strictly speaking, the thrust from a rocket engine is pushing against the rocket.
We tend to think that we need to push against something because that’s how our bodies work when we walk or swim, but a sturdy balloon deflating in a vacuum would fly around just as easily as one flying around in the air (more easily, in fact, because there’s no wind resistance)
When a rocket fires its engines, it is “pushing” against the gases that exit. Newton’s 3rd law, which you seem to cite, states that every action has an equal and opposite reaction. Think of this as forces being mutual. Multiply a force by the amount of time that force is applied, and you’ve got the change in momentum. So a rocket “pushes” its exhaust gases out very quickly out the back, applying force over some time. Those exhaust gases have momentum in that backwards direction. The universe conserves momentum, however, so the rocket has to have equal and opposite momentum (so the momenta cancel each other out). This moves the rocket in the forward direction. Since the rocket is more massive than the exhaust gases, the rocket will move more slowly but have the same momentum (which is equal to the product of mass and velocity).
Wrong. Newton’s third law states that for every action there is an equal and opposite reaction. Imagine floating in space. If you throw a baseball you will drift in the opposite direction, but not as fast as your mass is a lot greater than the baseball. But the energy imparted to the ball, will be equal to the energy applied to your body.
Now instead of a baseball, use one of those huge medicine balls they made us toss around in gym class. The ball will move slower away from you, but you will move faster than you did with the baseball. The energy of the medicine ball is equal to the energy of your body.
Now instead of balls, we use a chemical explosion, that converts liquid fuel to fast expanding gas, you need to throw away a lot of gas very fast to move the heavy rocket quickly.
Despite a couple other responses you correctly identified that there are two objects involved in any force (Newton’s third law).
In this case you fell into the common blind spot of forgetting that a gas is an object. The rocket engine pushes the exhaust backwards hard, and that pushes the rocket forward.
And this is what it does the entire time, even down here in the atmosphere. The air here is actually slowing the rocket, as it’s in front of the rocket and has to be pushed out of the way for the rocket to ascend.
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