understand that, while momentum of a system is always conserved in interactions between objects, some change in kinetic energy may take place

Linear Momentum and its Conservation

Definition of Linear Momentum

Linear momentum \$\mathbf{p}\$ of a particle of mass \$m\$ moving with velocity \$\mathbf{v}\$ is defined as

\$\mathbf{p}=m\mathbf{v}.\$

Momentum of a System

For a system of \$N\$ particles, the total momentum is the vector sum

\$\mathbf{P} = \sum{i=1}^{N} mi \mathbf{v}_i.\$

Newton’s Second Law in Momentum Form

Newton’s second law can be written as

\$\mathbf{F}_{\text{net}} = \frac{d\mathbf{p}}{dt}.\$

Conservation Principle

If the net external force on a closed system is zero, then

\$\frac{d\mathbf{P}}{dt}=0 \quad\Longrightarrow\quad \mathbf{P}{\text{initial}} = \mathbf{P}{\text{final}}.\$

This is the law of conservation of linear momentum.

Elastic vs Inelastic Collisions

Collisions are classified according to the change in kinetic energy:

• Elastic collision: both momentum and kinetic energy are conserved.
• Inelastic collision: momentum is conserved but kinetic energy is not; some kinetic energy is transformed into other forms (heat, deformation, sound, etc.).
• Perfectly inelastic collision: the maximum possible kinetic energy loss; the colliding bodies stick together after impact.

Quantitative Example

Derivation of Final \cdot elocities in a One‑Dimensional Elastic Collision

Consider two masses \$m1\$ and \$m2\$ moving along a line with initial velocities \$u1\$ and \$u2\$. Using conservation of momentum and kinetic energy:

\$m1 u1 + m2 u2 = m1 v1 + m2 v2,\$

\$\frac{1}{2}m1 u1^2 + \frac{1}{2}m2 u2^2 = \frac{1}{2}m1 v1^2 + \frac{1}{2}m2 v2^2.\$

Solving these equations gives

\$v1 = \frac{(m1 - m2)u1 + 2m2 u2}{m1 + m2},\$

\$v2 = \frac{(m2 - m1)u2 + 2m1 u1}{m1 + m2}.\$

Derivation for a Perfectly Inelastic Collision

If the two bodies stick together after impact, let the common final velocity be \$v\$. Conservation of momentum gives

\$(m1 + m2) v = m1 u1 + m2 u2 \quad\Longrightarrow\quad v = \frac{m1 u1 + m2 u2}{m1 + m2}.\$

The loss of kinetic energy is

\$\Delta K = \frac{1}{2}\frac{m1 m2}{m1 + m2}(u1 - u2)^2,\$

which is always non‑negative, showing that kinetic energy decreases (or remains unchanged) in an inelastic collision.

Key Points to Remember

1. Momentum is a vector quantity; both magnitude and direction must be considered.
2. In a closed system, total momentum is conserved regardless of the nature of the interaction.
3. Kinetic energy is conserved only in elastic collisions; otherwise it is transformed into other forms.
4. Use the conservation equations appropriate to the problem: momentum always, kinetic energy only when the collision is known to be elastic.
5. Remember to check the reference frame; momentum conservation holds in any inertial frame.