Conservation of Momentum
Momentum is the product of an object's mass and velocity, and it is one of the most powerful conserved quantities in physics. The law of conservation of momentum states that in an isolated system — one where no net external force acts — the total momentum before any interaction equals the total momentum afterwards. This principle holds true whether we are studying a pair of ice skaters pushing apart, a rocket launching into space, or subatomic particles colliding in an accelerator. It is not merely a useful rule of thumb; it is a direct consequence of Newton's third law of motion. An isolated system is the key requirement. In practice, 'isolated' means external forces are either absent or negligible compared with the internal forces during the interaction. A collision between two billiard balls on a frictionless table is a good approximation: gravity and the normal force cancel vertically, and friction is small enough to ignore during the brief collision. The enormous internal forces the balls exert on each other are equal and opposite, so they cancel when you consider the system as a whole, leaving the total momentum unchanged. Conservation of momentum applies in all directions independently, because momentum is a vector quantity. A two-dimensional collision must conserve the x-component and y-component of momentum separately. This vector nature makes conservation of momentum far more powerful than scalar energy conservation alone, allowing physicists to deduce the outcome of complex collisions even without knowing the detailed forces involved.
- Define the System
- Why Internal Forces Cancel
- Write the Conservation Equation
- Types of Collision
- Explosions and Recoil
Define the System
Identify every object involved in the interaction. Draw a boundary around them — this is your system. List all forces: those between objects inside the boundary are internal forces; those from outside are external forces. Only when net external force is zero (or negligible) is the system truly isolated and momentum conserved.
Why Internal Forces Cancel
Newton's third law tells us that every internal force has an equal and opposite reaction force also inside the system. When you add up all internal forces across the whole system, they cancel in pairs, giving zero net internal force. Because net force equals the rate of change of momentum (F = dp/dt), zero net force means momentum does not change — it is conserved.
Write the Conservation Equation
Label objects 1 and 2 with masses m₁ and m₂. Before the collision their velocities are u₁ and u₂; after the collision they are v₁ and v₂. Conservation of momentum gives: m₁u₁ + m₂u₂ = m₁v₁ + m₂v₂. This one equation (or one per direction in 2D) is your main tool. Notice it makes no assumption about whether kinetic energy is conserved — it works for elastic and inelastic collisions alike.
Types of Collision
In an elastic collision, both momentum and kinetic energy are conserved — ideal gas molecules and subatomic particles approximate this. In a perfectly inelastic collision, the objects stick together after impact, sharing a common final velocity; momentum is still conserved but kinetic energy is not (it is converted to heat, sound, or deformation). Most everyday collisions fall between these extremes, conserving momentum but losing some kinetic energy.
Explosions and Recoil
Conservation of momentum also governs situations where a system initially at rest flies apart — an explosion or a gun firing a bullet. If total initial momentum is zero, total final momentum must also be zero. The recoil velocity of the gun is therefore exactly equal and opposite (scaled by mass ratio) to the bullet's velocity. Rockets work on this principle: ejecting mass backwards at high speed gives the rocket forward momentum.
The Key Equation
- m_1,\, m_2 — Masses of objects 1 and 2 (kg): The masses of the two interacting objects; these remain constant in classical mechanics.
- \mathbf{u}_1,\, \mathbf{u}_2 — Initial velocities (m s⁻¹): Vector velocities of objects 1 and 2 immediately before the collision. Direction matters — choose a positive direction consistently.
- \mathbf{v}_1,\, \mathbf{v}_2 — Final velocities (m s⁻¹): Vector velocities of objects 1 and 2 immediately after the collision.
Worked Example
Problem. A 2.0 kg trolley moving at 3.0 m s⁻¹ to the right collides with a stationary 1.0 kg trolley on a frictionless track. After the collision the 2.0 kg trolley moves at 1.0 m s⁻¹ to the right. Find the velocity of the 1.0 kg trolley after the collision.
Answer. The 1.0 kg trolley moves at 4.0 m s⁻¹ to the right after the collision.
Origins · 1 minJohn Wallis and how this was discovered
John Wallis was one of the first scientists to state the law of conservation of momentum clearly and mathematically. In 1668, commissioned by the Royal Society, he published a paper on the collision of bodies in which he correctly described how the total momentum of colliding objects is preserved — predicting outcomes for both elastic and inelastic impacts. His work, alongside that of Christopher Wren and Christiaan Huygens published the same year, established collision theory as a rigorous science and laid a key foundation for Newton's later synthesis in the Principia. Wallis was also one of the greatest mathematicians of his era — he invented the symbol ∞ for infinity, which we still use today.
Related Concepts
- Momentum & Impulse — Impulse (force × time) equals the change in momentum of an object, directly linking Newton's second law to the momentum framework used in conservation.
- Newton's Third Law — Conservation of momentum is a direct mathematical consequence of Newton's third law — equal and opposite internal forces cause the momenta of interacting objects to change by equal and opposite amounts.
- Elastic & Inelastic Collisions — Classifying collisions as elastic or inelastic determines whether kinetic energy is also conserved alongside momentum, completing the analysis of any collision problem.
- Newton's Second Law — Newton's second law (F = dp/dt) shows that zero net external force means zero rate of change of momentum — the mathematical basis for why isolated systems conserve momentum.