Unit 4: Linear Momentum

Momentum

Momentum is a vector quantity, defined as:

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

• Momentum has no special unit name: $$\mathbf{p} = m\mathbf{v} \Rightarrow kg \cdot \frac{m}{s}$$

Conservation of Momentum

$$\sum \mathbf{p}_i = \sum \mathbf{p}_f$$

• Momentum is conserved during all **collisions and explosions**.
• For **two-dimensional collisions**, use separate equations for each axis: $$\sum \mathbf{p}_{xi} = \sum \mathbf{p}_{xf} \quad \text{and} \quad \sum \mathbf{p}_{yi} = \sum \mathbf{p}_{yf}$$

Types of Collisions

• Many real-world collisions are **inelastic** (somewhere between elastic and perfectly inelastic).
• Inelastic collisions: Objects bounce off each other, **momentum is conserved** but **kinetic energy is not**.
• Elastic and perfectly inelastic collisions represent **ideal extremes**.

Newton’s Second Law in Terms of Momentum

$$\sum \mathbf{F} = m \mathbf{a} = m \left(\frac{\Delta \mathbf{v}}{\Delta t} \right) = m \left(\frac{\mathbf{v}_f - \mathbf{v}_i}{\Delta t} \right)$$

Since momentum is defined as $\mathbf{p} = m\mathbf{v}$, we can rewrite it as:

$$\sum \mathbf{F} = \frac{m\mathbf{v}_f - m\mathbf{v}_i}{\Delta t} = \frac{\mathbf{p}_f - \mathbf{p}_i}{\Delta t} = \frac{\Delta \mathbf{p}}{\Delta t}$$

This gives the impulse equation:

$$\Delta \mathbf{p} = \sum \mathbf{F} \Delta t = \mathbf{J} = \text{Impulse}$$

Impulse Approximation

• For a very short time period, the force can be treated as an **average impact force**:

$$\sum \mathbf{F} \approx \mathbf{F}_{\text{impact}} \Rightarrow \Delta \mathbf{p} = \mathbf{F}_{\text{impact}} \Delta t = \mathbf{J}$$

• On a **Force vs. Time** graph, the **area under the curve** represents **impulse**.
• Units for impulse: $$\Delta \mathbf{p} = \mathbf{F}_{\text{impact}} \Delta t \Rightarrow N \cdot s = kg \cdot \frac{m}{s}$$