Why is the total momentum of an isolated system conserved, and how is this used to analyze collisions?
State the law of conservation of momentum, explain it using Newton's third law, and apply it to collisions and explosions where the total momentum before equals the total momentum after.
A Regents Physics answer on conservation of momentum: why total momentum is conserved in an isolated system, how Newton's third law explains it, and how to solve collision and explosion problems with total momentum before equal to total momentum after, with worked examples.
Reviewed by: AI editorial process; not yet individually human-reviewed
Have a quick question? Jump to the Q&A page
Jump to a section
What this topic is asking
Conservation of momentum is one of the two great conservation laws of Regents mechanics (the other is energy). The Physical Setting/Physics course asks you to state the law, explain it through Newton's third law, and apply it to collisions (objects coming together) and explosions or recoils (objects pushing apart). The recurring exam task is to set the total momentum before an interaction equal to the total momentum after, and solve for an unknown velocity.
The law of conservation of momentum
An "isolated system" is one where outside forces either do not act or cancel, so the only forces are the internal ones between the objects. In practice the Regents treats brief collisions and explosions as isolated, because the internal forces during the brief interaction dwarf any external ones (like friction) acting in that instant.
Why momentum is conserved: Newton's third law
This explanation is often worth a mark in itself. It connects the topic to Newton's third law and momentum and impulse, and it makes clear why the law needs no external force.
Applying the law to collisions
For a collision between two objects, write
choosing a positive direction and giving each velocity a sign. Two cases are common:
- Perfectly inelastic: the objects stick together and move off with a common velocity. The right side becomes .
- Elastic or general: the objects separate with different velocities; momentum is still conserved, and you solve for the unknown velocity.
In a perfectly inelastic collision, momentum is conserved but kinetic energy is not (some becomes heat, sound or deformation), a distinction explored in energy and its conservation.
Applying the law to explosions and recoils
In an explosion or recoil, objects initially together push apart. If the system starts at rest, the total momentum is zero before, so it must be zero after: the fragments carry equal and opposite momenta. A skater throwing a ball recoils backward; a gun recoils when a bullet is fired; a rocket moves forward as exhaust is expelled. The equation is the same conservation statement, often with a total of zero.
Reference Tables note
Conservation of momentum is not printed as an equation on the Reference Tables; the tables give only and . You must therefore recall the law and write "total momentum before equals total momentum after" yourself. This is one of the few relationships the Regents expects you to supply from memory rather than read from the booklet.
Try this
Q1. State the law of conservation of momentum. [2 points]
- Cue. In an isolated system, the total momentum before an interaction equals the total momentum after.
Q2. A kg ball at m/s strikes a stationary kg ball and stops; they do not stick. State the velocity of the second ball. [1 point]
- Cue. By conservation, the second ball moves off at m/s (the first ball's momentum is transferred entirely).
Exam-style practice questions
Practice questions written in the style of NYSED exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.
Regents (style)3 marksPart C (extended response). A kg cart moving at m/s collides with a stationary kg cart, and the two stick together. (a) State the law of conservation of momentum. (b) Calculate the common velocity of the carts after the collision. Show all work.Show worked answer β
A 3-point Part C perfectly inelastic collision.
(a) Law (1 point): in an isolated system (no external net force) the total momentum before an interaction equals the total momentum after.
(b) Velocity after (2 points): total momentum before kg m/s. After, the combined kg moves at : , so m/s in the original direction.
Markers reward writing total momentum before equal to total momentum after and solving for the common velocity. A common error is to conserve velocity or kinetic energy instead of momentum.
Regents (style)2 marksPart B-2 (constructed response). A kg skater, initially at rest, throws a kg ball east at m/s. Calculate the skater's recoil velocity. Show the equation, substitution and answer.Show worked answer β
A 2-point constructed-response recoil (explosion-type) problem. Total momentum before is zero, so the momenta after must be equal and opposite.
Equation: total momentum before total momentum after, so .
Substitution: , so .
Answer: m/s, that is m/s west (opposite to the ball).
Markers reward setting the total momentum to zero before and after and solving for the recoil velocity with its (opposite) direction.
Related dot points
- Define momentum as , define impulse as , and apply the impulse-momentum relationship to calculate force, time or change in momentum.
A Regents Physics answer on momentum and impulse: momentum as mass times velocity, impulse as force times time, and the impulse-momentum relationship from the Reference Tables, with applications to collisions and safety, plus worked examples.
- State Newton's third law, identify action-reaction force pairs, and explain why the two forces in a pair act on different objects and therefore do not cancel.
A Regents Physics answer on Newton's third law: that forces occur in equal and opposite pairs, how to identify an action-reaction pair, why the pair acts on different objects, and why this means the forces never cancel, with worked examples and Reference-Table notes.
- Define kinetic energy, gravitational potential energy and elastic potential energy, and apply the conservation of energy to systems with and without friction, recognizing friction transfers mechanical energy to internal (thermal) energy.
A Regents Physics answer on mechanical energy and its conservation: kinetic energy, gravitational and elastic potential energy, the conservation of energy with and without friction, and how friction transfers energy to heat, using the Reference-Table equations, with worked examples.
- State and apply Newton's second law, , to calculate net force, mass or acceleration, and analyze situations with several forces by finding the net force first.
A Regents Physics answer on Newton's second law: the relationship between net force, mass and acceleration, why acceleration is proportional to net force and inversely proportional to mass, and how to solve multi-force problems, with worked examples and Reference-Table notes.
- Define work as for a force along the displacement, relate work to the energy transferred, and define power as the rate of doing work, .
A Regents Physics answer on work and power: what work is and when a force does it, the link between work and energy transfer, and power as the rate of doing work, using the Reference-Table equations , and , with worked examples.
Sources & how we know this
- Reference Tables for Physical Setting/Physics β NYSED (2006)
- Physical Setting/Physics Core Curriculum β NYSED (2010)