Is Momentum Conserved If A Spring Is In The Collision

Is Momentum Conserved When a Spring Is Involved in a Collision?

The principle of conservation of momentum is one of the most powerful and reliable laws in physics, stating that the total momentum of an isolated system remains constant if no net external force acts upon it. This fundamental rule governs everything from billiard ball collisions to the motion of galaxies. However, the introduction of a spring—a device that stores and releases energy—often leads to a common and understandable point of confusion. When two objects collide and a spring between them compresses or expands, does this internal mechanism violate the sacred law of momentum conservation? The definitive answer is no. Momentum remains conserved in such interactions, provided the system of the two objects (and the spring) is truly isolated from external forces. The spring, while it transforms kinetic energy into potential energy and back again, exerts internal forces that always come in action-reaction pairs, perfectly canceling out within the system. The apparent complexity arises from confusing the conservation of momentum with the conservation of kinetic energy, which is a separate and often violated principle in real-world collisions.

The Unshakable Foundation: What "Conserved" Really Means

To understand why a spring doesn’t break the rule, we must return to the bedrock definition. Momentum (p) is the product of an object's mass and its velocity (p = m*v). For a system of multiple objects, the total momentum is the vector sum of all individual momenta. The Law of Conservation of Momentum is a direct consequence of Newton's Third Law of Motion: for every force, there is an equal and opposite force.

Consider any two objects, A and B, interacting. When A exerts a force on B (F_AB), B simultaneously exerts a force on A (F_BA) that is equal in magnitude and opposite in direction (F_AB = -F_BA). The impulse (force multiplied by time) delivered by A to B changes B’s momentum by a certain amount. Simultaneously, the impulse from B to A changes A’s momentum by the exact opposite amount. These changes are equal and opposite, meaning the total momentum change of the A-B system is zero. Therefore, the total momentum before the interaction equals the total momentum after. This logic holds true regardless of the nature of the internal forces—whether they are contact forces, gravitational pulls, or the elastic forces from a compressed spring. The key is that these forces are internal to the defined system.

The Spring’s Role: An Internal Force Machine

A spring is a perfect example of an internal force generator. When compressed or stretched, it exerts a restoring force described by Hooke's Law (F = -kx), where k is the spring constant and x is the displacement from equilibrium. This force is not an external push from the outside world; it is a force arising from the interaction between the two objects connected to the spring.

Imagine two masses, m₁ and m₂, on a frictionless horizontal surface. A spring is compressed between them and then released. As the spring expands:

1. It pushes backward on m₁ and forward on m₂.
2. The force on m₁ is F, and the force on m₂ is -F (equal in magnitude, opposite in direction).
3. m₁ gains momentum in the negative direction, while m₂ gains an equal amount of momentum in the positive direction.
4. The sum of their momenta remains exactly what it was when the spring was compressed (which was zero if they started at rest relative to each other).

The spring itself, if considered part of the system, has negligible mass and thus negligible momentum. Its role is purely to mediate the transfer of momentum between the two masses. It does not create or destroy net momentum; it only redistributes it between the objects it connects. The energy story is different: the spring stores kinetic energy as elastic potential energy, which can later be converted back. This exchange can make the kinetic energy of the system non-constant, but the total momentum is locked in, a direct result of the paired internal forces.

A Classic Example: The Spring-Loaded Plunger

This scenario is a staple in physics textbooks for a reason. Two carts on an air track (to minimize friction, an external force) face each other. A spring-loaded plunger on one cart is pressed against the other cart and then released.

• Initial State (before release): Both carts are at rest. Total momentum = 0.
• During Expansion: The spring exerts forces. Cart A is pushed left, Cart B is pushed right. Their velocities change in opposite directions.
• Final State (after spring fully expands): Both carts are moving apart. Cart A has momentum p_A to the left. Cart B has momentum p_B to the right.
• The Check: Because the forces were equal and opposite at every instant, |p_A| = |p_B|. Therefore, p_A + p_B = 0. Total momentum is still zero. It has been conserved.

The system’s kinetic energy, however, has increased from zero to the sum of the kinetic energies of both carts. This energy came from the potential energy stored in the compressed spring. If the spring were to later re-compress as the carts moved (in a perfectly elastic scenario on a closed track), that kinetic energy would convert back

If the spring were to later re‑compress as the carts moved (in a perfectly elastic scenario on a closed track), that kinetic energy would convert back into elastic potential energy, and the carts would slow, reverse direction, and eventually come to rest again with the spring maximally compressed. Throughout this oscillation the total momentum of the two‑cart‑plus‑spring system remains exactly zero at every instant, because the internal forces always occur in equal‑and‑opposite pairs. The center of mass of the system, which is defined by the weighted average of the positions of the masses, therefore stays fixed in space; it does not drift even though each cart individually speeds up and slows down.

This principle extends far beyond the simple spring‑loaded plunger. Consider a firecracker exploding inside a sealed, frictionless container. The explosive gases push outward on the container walls with forces that are internal to the system; the container recoils in the opposite direction with exactly the same momentum that the fragments acquire. The net momentum of the container + fragments + gases stays zero, even though the internal chemical potential energy has been transformed into a large amount of kinetic energy. Similarly, a rocket in deep space expels exhaust gases backward; the rocket gains forward momentum equal in magnitude to the backward momentum of the expelled mass. No external thrust is required—the momentum exchange is wholly internal, mediated by the interaction between the rocket and its exhaust.

In all of these cases the internal agent (spring, explosive, exhaust) merely redistributes momentum among the parts of the system. It can store and release energy, thereby changing the kinetic energy distribution, but it cannot alter the total momentum unless an external force acts on the system as a whole. The conservation of momentum thus emerges as a direct consequence of Newton’s third law applied to every internal interaction pair.

Conclusion:
Whether visualized as two carts pushing apart via a compressed spring, fragments flying from an internal explosion, or a rocket propelled by its exhaust, the underlying physics is the same: internal forces occur in equal‑and‑opposite pairs, guaranteeing that the vector sum of momenta of all constituent parts remains unchanged. Energy may be shuffled between kinetic, potential, and internal forms, but momentum is a conserved quantity that only changes when the system interacts with something outside its boundaries. This invariance makes momentum a powerful tool for analyzing a wide range of mechanical processes, from simple laboratory demonstrations to the propulsion of spacecraft.