Every physical phenomenon we observe—from the descent of a skydiver to the collision of billiard balls to the launch of a rocket into space—follows a set of unchanging physical laws. The details of how we analyse these motions may vary depending on the situation, but the underlying principles remain the same.
One such principle is the law of conservation of linear momentum. It seems to emerge naturally from Newton’s laws of motion, but its significance extends far beyond them. In fact, it holds true even in domains where Newton’s laws break down or simply do not apply—such as the quantum realm or in phenomena involving light. As Emmy Noether, the brilliant German mathematician, demonstrated, this law is a consequence of a profound symmetry of nature, so fundamental that we rarely pause to reflect on it.
Understanding momentum through billiards
Consider a billiard ball rolling across the smooth surface of a table. As it rolls, friction between the ball and the cloth slows it down until it eventually stops. What determines how far it will travel before halting? Clearly, the ball’s initial speed matters—the faster it moves, the farther it goes before friction overcomes it.
Now imagine the same ball but twice as heavy, moving at the same speed. Friction again has a harder time stopping it. Thus, both mass and speed determine how difficult it is to bring the ball to rest. Specifically, the product of mass and velocity is defined as the ball’s linear momentum.
A lighter ball moving very fast can have the same momentum as a heavier ball moving more slowly. This explains why a bullet is dangerous: it weighs only a few grams, but when fired from a gun it acquires enormous speed, and hence a very large momentum. Conversely, even a slowly rolling boulder can be equally dangerous because of its immense mass.
In simple terms, momentum tells us how hard it is to stop an object. It may be large either because of high speed, large mass, or both.
Newton’s laws and momentum
Newton’s laws tell us that changing an object’s momentum requires a force. The faster we want to change it, the larger the force must be. When a gun is fired, the explosion of gunpowder exerts a huge force on the bullet in a very short time, giving it a high speed and rapidly increasing its momentum. Before the shot, the bullet rests with zero momentum; afterward, it moves with nearly constant momentum until forces like air resistance or gravity act significantly.
Thus, Newton’s laws imply that in the absence of external forces, momentum cannot change—our first glimpse of the conservation principle.
Internal vs. external forces
But what if the bullet is viewed as a collection of tiny particles, each exerting forces on its neighbours? Could these internal forces alter the bullet’s momentum? The answer is no. According to Newton’s third law, every action has an equal and opposite reaction. Internal forces cancel out, leaving the net momentum of the whole object unchanged. Pushing on the inside of a car windshield from within won’t move the car—you need an external force.
This reasoning extends to systems of interacting objects. For example, before a gun is fired, both gun and bullet are at rest with zero total momentum. When the trigger is pulled, the bullet shoots forward, but the gun recoils backward. The small backward velocity of the heavy gun exactly balances the large forward velocity of the light bullet, keeping the system’s total momentum unchanged.
The same applies to colliding billiard balls: the momentum lost by one is gained by the other, so the total remains conserved.
Rockets and slipping on ice
This principle also explains rocket launches. Initially, the rocket and its fuel are at rest. When the engine fires, fuel is expelled downward at high speed. To conserve momentum, the rocket acquires an equal and opposite momentum upward. Its final speed depends both on the velocity of the exhaust gases and the rate at which fuel is ejected.
A similar idea applies if you were stranded on a frictionless frozen lake. You cannot walk to safety, but if you throw say your shoe sideways, you slide in the opposite direction. Repeating this with other objects gradually propels you towards land—the law of conservation of momentum in action.
Beyond Newton: the deeper origin
At first glance, momentum conservation looks like a consequence of Newton’s laws. But it is deeper. Even in domains where Newton’s framework fails—such as quantum mechanics or optics—the law holds. For instance, when a light pulse bounces off a mirror, it imparts momentum to it, just as the law predicts.
So why is momentum conserved at all? The answer lies in a fundamental property of nature: the laws of physics do not depend on where you are in space. If you perform an experiment in one location and then repeat it elsewhere under identical conditions, the results are the same. In technical terms, space is homogeneous.
Emmy Noether proved mathematically in 1915 that this symmetry of space is exactly what gives rise to the conservation of momentum. If space were not homogeneous—if the outcome of an experiment depended on its location—the conservation of momentum would fail.
A universal truth
Thus, the law of conservation of linear momentum is not just a convenient tool for solving physics problems. It is a fundamental truth about the universe, rooted in the very structure of space itself. From the collisions of air molecules as you breathe, to the launch of rockets, to the behavior of particles of light, this law quietly governs the motion of everything.
