Momentum: $\mathbf{p} = m\mathbf{v}$ (kg$\cdot$m/s).
Impulse: $\mathbf{J} = \mathbf{F}\Delta t = \Delta\mathbf{p}$ (N$\cdot$s = kg$\cdot$m/s).
For variable force: $\mathbf{J} = \displaystyle\int_{t_1}^{t_2}\mathbf{F}\,dt$.
In a system with no net external force:
$$\sum m_i\mathbf{v}_i = \text{constant}$$
Internal forces (between particles in the system) cancel by Newton’s third law.
Applies to: collisions, explosions, any interaction with no external horizontal force.
| Type | Momentum conserved? | KE conserved? | Example |
|---|---|---|---|
| Elastic | Yes | Yes | Ideal billiard balls |
| Inelastic | Yes | No (some lost) | Typical collision |
| Perfectly inelastic | Yes | No (maximum loss) | Objects stick together |
For masses $m_1, m_2$ with initial velocities $u_1, u_2$:
$$v_1 = \frac{(m_1-m_2)u_1 + 2m_2 u_2}{m_1+m_2}$$
$$v_2 = \frac{(m_2-m_1)u_2 + 2m_1 u_1}{m_1+m_2}$$
Special case: equal masses ($m_1 = m_2$): $v_1 = u_2$ and $v_2 = u_1$ (velocities swap).
Example 1 (Perfectly inelastic): A 3 kg ball ($u_1 = 6$ m/s) collides with a stationary 5 kg ball; they stick together.
$$3(6) + 5(0) = (3+5)v \Rightarrow v = 18/8 = 2.25 \text{ m/s}$$
KE lost $= \frac{1}{2}(3)(36) - \frac{1}{2}(8)(5.0625) = 54 - 20.25 = 33.75$ J.
Example 2 (Explosion): A 10 kg shell at rest explodes into a 4 kg fragment ($v_1 = 15$ m/s right) and a 6 kg fragment. Find $v_2$.
$\$0 = 4(15) + 6v_2 \Rightarrow v_2 = -10 \text{ m/s (left)}$$
Example 3 (2D collision): A 2 kg ball ($\mathbf{u}_1 = 5\mathbf{i}$) hits a stationary 3 kg ball. After collision: $\mathbf{v}_1 = 2\mathbf{i}+2\mathbf{j}$. Find $\mathbf{v}_2$.
$$2(5\mathbf{i}) = 2(2\mathbf{i}+2\mathbf{j})+3\mathbf{v}_2$$
$$10\mathbf{i} = 4\mathbf{i}+4\mathbf{j}+3\mathbf{v}_2$$
$$\mathbf{v}_2 = \frac{6\mathbf{i}-4\mathbf{j}}{3} = 2\mathbf{i}-\frac{4}{3}\mathbf{j} \text{ m/s}$$
$$e = \frac{v_2 - v_1}{u_1 - u_2} = \frac{\text{relative speed of separation}}{\text{relative speed of approach}}$$
KEY TAKEAWAY: Momentum is always conserved in collisions and explosions (no external force). Kinetic energy is only conserved in elastic collisions. Always verify conservation in both components for 2D problems.
EXAM TIP: For perfectly inelastic collisions, the combined object moves with one velocity. Use $\sum m_i u_i = (\sum m_i)v$ and solve for $v$.
COMMON MISTAKE: Checking conservation of kinetic energy when only conservation of momentum is needed. Unless the problem states the collision is elastic, do not assume KE is conserved.
