Equal & Opposite Reactions: Newton’s Third Law of Motion

Wallops flight facility, Virginia, national parks
A test rocket being launched at Wallops Flight Facility in Virginia.
Credit: NASA Wallops Flight Facility

Isaac Newton’s Second Law of Motion describes what happens when an external force acts upon a massive body at rest or in uniform linear motion. What happens to the body from which that external force is being applied? That situation is described by Newton’s Third Law of Motion. It states, “For every action, there is an equal and opposite reaction.” 

Forces always occur in pairs; when one body pushes against another, the second body pushes back just as hard. For example, when you push a cart, the cart pushes back against you; when you pull on a rope, the rope pulls back against you; and when gravity pulls you down against the ground, the ground pushes up against your feet. The simplified version of this phenomenon has been expressed as, “You cannot touch without being touched.”

If body A exerts a force F on body B, then body B exerts an equal and opposite force −back on body A. The mathematical expression for this is:


The subscript AB indicates that A exerts a force on B, and BA indicates that B exerts a force on A. The minus sign indicates that the forces are in opposite directions. Often FAB and FBA are referred to as the action force and the reaction force; however, the choice of which is which is completely arbitrary. 

If one object is much, much more massive than the other, particularly in the case of the first object being anchored to the Earth, virtually all of the acceleration is imparted to the second object, and the acceleration of the first object can be safely ignored. For instance, if you were to plant your feet and throw a baseball to the west, you would not have to consider that you actually caused the rotation of the Earth to speed up slightly while the ball was in the air. However, if you were standing on roller skates, and you threw a bowling ball forward, you would start moving backward at a noticeable speed. 

One might ask, “If the two forces are equal and opposite, why do they not cancel each other out?” Actually, in some cases they do. Consider a book resting on a table. The weight of the book pushes down on the table with a force mg, while the table pushes up on the book with an equal and opposite force. In this case, the forces cancel each other because the book does not accelerate. The reason for this is that both forces are acting on the same body, while Newton’s Third Law describes two different bodies acting on each other. 

Consider a horse and a cart. The horse pulls on the cart, and the cart pulls back on the horse. The two forces are equal and opposite, so why does the cart move at all? The reason is that the horse is also exerting a force on the ground, which is external to the horse–cart system, and the ground exerts a force back on the horse–cart system causing it to accelerate. 

How, then, can a rocket move through space if there is nothing for it to push against? When the fuel is ignited in the rocket nozzle, the gas expands rapidly in all directions. Some of it goes backwards and has no effect on the rocket; however, some if it goes forward and crashes into the back of the rocket exerting a force that causes the rocket to accelerate in the forward direction. This is why Newton’s Third Law is considered to be the fundamental principle of rocket science. 

Jim Lucas is a freelance writer and editor specializing in physics, astronomy and engineering. He is general manager of Lucas Technologies

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