Linear momentum is the product of an object's mass and velocity. The total linear momentum of a system remains constant when no resultant external force acts on it, or equivalently, when the net external impulse is zero.
For both SL and HL, momentum is defined by
where is mass in kilograms, is velocity in metres per second, and is momentum in . Momentum is a vector quantity, so its direction must be included; motion in the opposite direction is normally assigned a negative sign.
A resultant external force changes the total momentum of a system. Newton's second law can be written as
Therefore, if , then , so total momentum is conserved. During a collision or explosion, internal forces occur in equal and opposite pairs according to Newton's third law. Their impulses cancel when the whole system is considered.
| Situation | Is total momentum conserved? |
|---|---|
| Collision with negligible external impulse | Yes |
| Explosion of an initially stationary object | Yes; the vector sum remains zero |
| System experiencing a resultant external force | Not necessarily |
For example, a trolley moving at collides with a stationary trolley, and they stick together:
The initial and final total momentum are both .
This is an inelastic collision because the trolleys stick together. Their kinetic energy decreases because some energy is transferred to thermal energy, sound, and deformation, but the internal interaction cannot change the system's total momentum. This distinction is central to syllabus topic A.2.
A common misconception is that kinetic energy is always conserved. Momentum is conserved in every isolated collision, but kinetic energy is conserved only in an elastic collision.
In an IB response, define the system, choose a positive direction, write , and retain velocity signs. For an Explain question, link zero resultant external force to zero change in total momentum rather than merely stating the conservation law.
