Newtonian mechanics

Conservation of momentum

A system's total momentum remains constant when the net external force is zero or negligible.

During a brief interaction, two objects can exert large forces on each other, but these are internal forces of the system. If external forces provide negligible impulse, total momentum is the same before and after.

\vec P = \sum_i \vec p_i, \qquad \Delta\vec P = \vec J_{\mathrm{ext}}, \qquad \vec J_{\mathrm{ext}}\simeq0 \Rightarrow \vec P_i=\vec P_f

Definition: Condition for conservation

Total system momentum is conserved when external impulse over the interval is zero or negligible for the desired accuracy. A system may contain many objects; internal forces cancel in pairs by Newton's third law.

Set masses, directed velocities, and external impulse; compare total system momentum before and after.

Isolated systems and collisions

Collision time is often short, so external impulse may be small relative to internal impulse, but that assumption must be checked. Momentum conservation does not imply kinetic-energy conservation: in an inelastic collision, some mechanical energy becomes internal energy, sound, or deformation.

Example: Two carts

A 2 kg cart moving right at 3 m/s sticks to a stationary 1 kg cart. Neglect horizontal external impulse; find their common velocity.

Solution

Take right as positive: $p_i=2(3)+1(0)=6$ kg·m/s. Thus $v_f=p_i/(2+1)=2$ m/s to the right.

Quick check

The law follows from Newton’s second law for a system: the rate of change of total momentum equals the net external force. Define the system before writing the equation; if table friction is appreciable or the interaction lasts a long time, external impulse cannot be neglected. In a brief collision on a horizontal surface, gravity and the normal force nearly balance vertically, while horizontal friction impulse is often small.

In a one-dimensional problem, choose a positive direction first and represent opposite velocities with negative signs. For two bodies, $m_1u_1+m_2u_2=m_1v_1+m_2v_2$ is a scalar equation; in multiple dimensions write one equation per axis. Conservation applies to the system as a whole, not to each object separately. Always check external forces over the specific interval being studied.

In an isolated system, two objects have total momentum 8 kg·m/s before interaction. What is it afterward?

Which quantity is directly guaranteed to remain constant by momentum conservation?

References

Young, Freedman (2019). University Physics