3.3 Linear momentum and its conservation
- Syllabus
- 9702–2028–2029
- Topic
- 3.3
- Level
- AS
Learning objectives
3.3.1The principle of conservation of momentum• state the principle of conservation of momentum3.3.2The principle of conservation of momentum to solve simple problems• apply the principle of conservation of momentum to solve simple problems, including elastic and inelastic interactions between objects in both one and two dimensions (knowledge of the concept of coefficient of restitution is not required)3.3.3That, for an elastic collision, total kinetic energy is conserved and the• recall that, for an elastic collision, total kinetic energy is conserved and the relative speed of approach is equal to the relative speed of separation3.3.4That, while momentum of a system is always conserved in interactions between• understand that, while momentum of a system is always conserved in interactions between objects, some change in kinetic energy may take place
Total momentum stays constant in an isolated system
The total momentum of a system of interacting objects remains constant when no resultant external force acts on the system: total momentum before = total momentum after.
Define which objects belong to the system, choose positive directions and add their vector momenta. Internal interaction forces can change each object's momentum but produce equal and opposite changes in the system total.
An initially stationary firework has total momentum zero. After it explodes into two fragments, their momenta are equal in magnitude and opposite in direction, so the total remains zero.
Isolation applies to the chosen system, not to each object. Momentum is conserved in elastic and inelastic interactions; elasticity is a separate kinetic-energy condition.
Conserve momentum separately in each direction
| Interaction geometry | Momentum method |
|---|---|
| One dimension | choose one positive direction and use signed velocities in Σmu = Σmv |
| Two dimensions | resolve every momentum and conserve x-components and y-components in two separate equations |
A 2.0 kg cart moving at +3.0 m s⁻¹ sticks to a stationary 1.0 kg cart. Before: total p = 6.0 kg m s⁻¹. After: combined mass = 3.0 kg, so v = 6.0/3.0 = +2.0 m s⁻¹.
Use the same momentum method for collisions, sticking interactions, recoil and explosions. Whether the interaction is elastic or inelastic changes the kinetic-energy analysis, not the isolated-system momentum equation.
Conserve vector momentum, not speed or momentum magnitude. In two dimensions, one scalar equation cannot determine both directional components.
An elastic collision satisfies two kinetic conditions
| Condition for an elastic collision | Statement |
|---|---|
| Total kinetic energy | Σ½mu² = Σ½mv² |
| Relative speed in one dimension | relative speed of approach = relative speed of separation |
For an isolated collision, momentum is also conserved. Use momentum together with either elastic condition to solve unknown velocities; do not replace signed velocities with speeds in the momentum equation.
In a head-on elastic collision of identical balls where one is initially stationary, the moving ball can stop and the other leave with its speed. Total momentum and kinetic energy are unchanged, and approach speed equals separation speed.
Momentum conservation alone does not prove that a collision is elastic. The kinetic-energy or relative-speed condition must also hold.
Momentum can stay constant while kinetic energy changes
| Isolated interaction | Total momentum | Total kinetic energy |
|---|---|---|
| Elastic collision | conserved | conserved |
| Inelastic collision, including sticking | conserved | decreases as energy transfers to internal, sound or deformation stores |
| Explosion/recoil | conserved | may increase as stored energy becomes kinetic energy |
Momentum conservation follows from the absence of a resultant external force on the system. It does not require kinetic energy to remain constant. Total energy is still conserved when kinetic energy changes form.
‘Kinetic energy is not conserved’ does not mean energy disappeared, and it does not invalidate momentum conservation for the isolated system.