# Linear Momentum ## What is Linear Momentum? - When any object that has **mass** is moving with a **velocity**, it possesses **momentum** - **Linear momentum** refers specifically to motion along a straight line (one-dimensional motion) - Without an external resultant force acting on the system, the linear momentum of a moving object stays **constant**

Momentum is defined as the product of an object's mass and velocity: $p = mv$

## The Momentum Equation - The defining equation for momentum is: $$p = mv$$ - Where: - $p$ = momentum, measured in kg m s$^{-1}$ - $m$ = mass, measured in kg - $v$ = velocity, measured in m s$^{-1}$ ### Visualizing the Momentum-Velocity Relationship - Since momentum is directly proportional to velocity, the relationship is **linear** - For a fixed mass, doubling the velocity doubles the momentum - The graph below shows how momentum varies with velocity for an object of mass 2.0 kg:

- The gradient of this line equals the mass of the object - A more massive object would have a steeper gradient ## Vector Nature of Momentum - Momentum is a **vector quantity** — it has both magnitude and direction - The direction of momentum is always the same as the direction of the velocity - Usually, we define the **initial direction of travel** as the positive direction - Velocities in this direction are positive - Velocities in the opposite direction are negative - Similarly, momentum values follow the same sign convention

When choosing a positive direction, stay consistent throughout your calculations to avoid sign errors

## Understanding Momentum Direction - Consider a football being kicked towards a goal, then bouncing back off the post: - **Before impact:** The ball travels towards the post with positive velocity (e.g., $+12$ m s$^{-1}$) - Its momentum is positive: $p = m \times (+12)$ kg m s$^{-1}$ - **After impact:** The ball rebounds away from the post with velocity in the opposite direction (e.g., $-8$ m s$^{-1}$) - Its momentum is now negative: $p = m \times (-8)$ kg m s$^{-1}$ - The change in direction is captured by the negative sign in the velocity, which also makes the momentum negative ## Unit Conversions for Momentum - The SI unit for momentum is **kg m s$^{-1}$** (kilogram metre per second) - Ensure all quantities are in SI units before substituting into $p = mv$: - **Mass:** If given in grams (g), convert to kilograms by dividing by 1000 - Example: 250 g = 0.25 kg - **Velocity:** If given in km s$^{-1}$, convert to m s$^{-1}$ by multiplying by 1000 - Example: 2.5 km s$^{-1}$ = 2500 m s$^{-1}$

Always convert mass from grams to kilograms before calculating momentum

## Worked Example: Comparing Momenta **Question:** A hockey puck of mass 160 g slides across ice to the right at 8.0 m s$^{-1}$. A bowling ball of mass 5.0 kg rolls to the right at a speed of 0.25 m s$^{-1}$. Which object has the greater momentum? **Answer:** **Step 1: List known quantities** - Hockey puck: - Mass, $m_{\text{puck}} = 160 \text{ g} = 0.16 \text{ kg}$ - Velocity, $v_{\text{puck}} = 8.0 \text{ m s}^{-1}$ - Bowling ball: - Mass, $m_{\text{ball}} = 5.0 \text{ kg}$ - Velocity, $v_{\text{ball}} = 0.25 \text{ m s}^{-1}$ **Step 2: Calculate momentum of the hockey puck** $$p_{\text{puck}} = m_{\text{puck}} \times v_{\text{puck}}$$ $$p_{\text{puck}} = 0.16 \times 8.0 = 1.28 \text{ kg m s}^{-1}$$ **Step 3: Calculate momentum of the bowling ball** $$p_{\text{ball}} = m_{\text{ball}} \times v_{\text{ball}}$$ $$p_{\text{ball}} = 5.0 \times 0.25 = 1.25 \text{ kg m s}^{-1}$$ **Step 4: Compare** - $p_{\text{puck}} = 1.28$ kg m s$^{-1}$ - $p_{\text{ball}} = 1.25$ kg m s$^{-1}$ **The hockey puck has the greater momentum**, even though the bowling ball is much heavier. ### Visual Comparison The bar chart below compares the momenta of both objects:

- This demonstrates that momentum depends on **both** mass and velocity - A lighter object moving quickly can have more momentum than a heavier object moving slowly ## Key Points Summary | **Concept** | **Details** | |-------------|-------------| | **Definition** | Momentum = mass × velocity | | **Equation** | $p = mv$ | | **SI Unit** | kg m s$^{-1}$ | | **Type** | Vector quantity | | **Direction** | Same as velocity direction | | **Sign convention** | Positive in initial direction of travel; negative in opposite direction | ## Sign Convention Reminder - You can choose any direction as positive, but be **consistent** - Typically, the direction of initial motion is taken as positive - Objects moving in the opposite direction have negative velocity and negative momentum