This topic covers the collective behaviour of multiple particles, energy methods, and momentum conservation.
For a system of particles with masses $m_i$ at positions $\mathbf{r}i$:
$$\mathbf{r}{\text{cm}} = \frac{\sum m_i \mathbf{r}_i}{\sum m_i} = \frac{\sum m_i \mathbf{r}_i}{M}$$
Example 1: Two particles: 3 kg at $(1,2)$ and 5 kg at $(5,6)$.
$$\mathbf{r}_{\text{cm}} = \frac{3(1,2)+5(5,6)}{8} = \frac{(3,6)+(25,30)}{8} = \frac{(28,36)}{8} = (3.5,\ 4.5)$$
Work done by force $\mathbf{F}$ along displacement $\mathbf{s}$:
$$W = \mathbf{F}\cdot\mathbf{s} = Fs\cos\theta$$
Kinetic energy: $KE = \dfrac{1}{2}mv^2$.
Gravitational potential energy: $GPE = mgh$ (height $h$ above reference).
Work-energy theorem: $W_{\text{net}} = \Delta KE$.
Conservation of energy (no non-conservative forces):
$$KE_1 + PE_1 = KE_2 + PE_2$$
Example 2: A 2 kg ball is released from rest at height 5 m. Find its speed at the bottom.
$$mgh = \tfrac{1}{2}mv^2 \Rightarrow v = \sqrt{2gh} = \sqrt{2\times9.8\times5} = \sqrt{98} \approx 9.9 \text{ m/s}$$
Linear momentum: $\mathbf{p} = m\mathbf{v}$ (kg$\cdot$m/s).
Impulse-momentum theorem: $\mathbf{J} = \mathbf{F}\Delta t = \Delta\mathbf{p}$.
Conservation of momentum (no external forces):
$$\sum m_i\mathbf{v}_i = \text{constant}$$
Example 3: Two particles collide. $m_1 = 2$ kg, $u_1 = 5$ m/s, $m_2 = 3$ kg, $u_2 = -1$ m/s. Find $v_1, v_2$ after elastic collision.
Conservation of momentum: $2(5)+3(-1) = 2v_1+3v_2 \Rightarrow 7 = 2v_1+3v_2$.
Conservation of kinetic energy (elastic):
$\tfrac{1}{2}(2)(25)+\tfrac{1}{2}(3)(1) = \tfrac{1}{2}(2)v_1^2+\tfrac{1}{2}(3)v_2^2 \Rightarrow 53 = 2v_1^2+3v_2^2$.
Or use elastic collision formulas:
$$v_1 = \frac{(m_1-m_2)u_1+2m_2 u_2}{m_1+m_2} = \frac{-5+(-6)}{5} = \frac{-11}{5} = -2.2 \text{ m/s}$$
$$v_2 = \frac{(m_2-m_1)u_2+2m_1 u_1}{m_1+m_2} = \frac{(-3)+20}{5} = 3.4 \text{ m/s}$$
KEY TAKEAWAY: Energy methods (work-energy theorem, conservation of energy) avoid the need to find forces explicitly. Momentum is always conserved in a closed system; kinetic energy is only conserved in elastic collisions.
EXAM TIP: For collision problems, always check whether to use conservation of momentum alone (inelastic) or both momentum and kinetic energy (elastic).
COMMON MISTAKE: Using the work formula $W = Fs$ (scalar) when the force and displacement are not in the same direction. Always use $W = \mathbf{F}\cdot\mathbf{s} = Fs\cos\theta$.
