Momentum and Impulse Notes

Learning Intentions and Success Criteria

Learning Intentions:
- Introduce the concept of momentum and its relation to force, acceleration, and energy.
Success Criteria:
- Determine proportionalities between momentum, mass, and velocity.
- Calculate momentum and the change in momentum of objects.

Momentum - Chapter 9

Linear Momentum

Linear refers to a line.
Angular momentum (momentum in a circle) will be covered later.
Momentum is challenging to define.

Linear Momentum Defined

Linear momentum measures how hard it is to stop a moving object; it indicates how strongly something is moving in a direction.
Example:
- Which is harder to stop: a freight train or a bicycle, both moving at 1 m/s?
- The train is harder to stop because it has more inertia.

Factors Affecting Momentum

How to make a bicycle as hard to stop as a train:
- Increase its mass to be the same as the train.
- Increase its speed immensely.

Linear Momentum: Mass and Velocity

Both inertia and velocity play a role in momentum.
Equation: $p = mv$ , where:
- $p$ represents momentum.
- $m$ represents mass.
- $v$ represents velocity.
Linear momentum is a vector quantity.
SI Unit: kg⋅m/s

Linear Momentum: Proportionality

Formula: $p = mv$
Directly Proportional:
- $p$ and $m$
- $p$ and $v$
Inversely Proportional:
- $m$ and $v$ (when $p$ is constant)

Linear Momentum Problem

Problem: What is the momentum of a 100 kg linebacker running at -3.00 m/s?
Solution:
- Given: $m = 100 \text{ kg}$ , $v = -3.00 \text{ m/s}$
- $p = m \times v$
- $p = 100 \text{ kg} \times -3.00 \text{ m/s}$
- $p = -300 \text{ kg m/s}$

Momentum - The Mathematical Center of Mechanics

Review on Graphical Analysis

Read the graph, noting the axes, and interpret what the graph is conveying.
Determine if the slope of the graph represents something significant, such as instantaneous acceleration or instantaneous force.
Assess if the area under the curve represents something meaningful, like the change in displacement or kinetic energy.

Momentum vs. Time

Reading the Graph:
- The object's momentum increases linearly.
- The object's speed increases.
- The object has acceleration.
- The object must have a net force.

Momentum vs. Time: Slope

Slope:
- Dimensional Analysis:
  - $slope = \frac{\Delta P}{\Delta t} = \frac{\text{kg m/s}}{\text{s}} = \text{kg m/s}^2$
  - $\text{kg m/s}^2$ is equivalent to a Newton (N).
- The slope of this graph represents the average force acting on the object between two time points.

Momentum vs. Time: Slope Explained

Slope:
- Variable Analysis:
  - $slope = \frac{\Delta P}{\Delta t} = \frac{m \Delta v}{\Delta t} = m a$
  - $F = m a$
- The slope of this graph represents the average force acting on the object between two time points.

Momentum vs. Time: Area

Area:
- Not particularly meaningful in this context.
- Units would be kg m.
- Mathematically, it would be $p \times t$ .

Momentum vs. Velocity

Reading the Graph:
- The object's momentum increases linearly.
- As the object's speed increases, its momentum increases.

Momentum vs. Velocity: Slope

Slope:
- Variable Analysis:
  - $\text{slope} = \frac{\Delta P}{\Delta v} = \frac{m \Delta v}{\Delta v} = m$
- Slope = mass of the object.

Momentum vs. Velocity: Area

Area under the curve:
- $\text{area} = \frac{1}{2} b h = \frac{1}{2} \Delta v \Delta p$
- $= \frac{1}{2} \Delta v \cdot m \Delta v = \frac{1}{2} m(\Delta v)^2$
- The area represents the change in the kinetic energy of the object from $v<em>0$ to $v</em>f$ .

Thinking Question: Net Force and Acceleration

Net force causes acceleration. but what else is needed to truly cause observed accelerations?
Time: No change occurs if $\Delta t = 0 \text{ s}$ .

Thinking Question: Force and Impact

If you punch a pillow with a force of 100 N, will it hurt as much as punching a wall with 100 N?
Justify your answer.

Force and Time

Unlike paper, the pillow can hit back with the same force, but it doesn't.
Force AND time play a role in why a pillow feels soft.

Learning Intentions and Success Criteria (Impulse)

Learning Intentions:
- Introduce the concept of impulse and its relation to force, acceleration, energy, and time.
Success Criteria:
- Determine proportionalities between momentum, mass, and velocity, force, and time.
- Describe Newton’s 3 laws of motion in terms of impulse.
- Calculate change in momentum, average force, and time of contact from word problems AND graphs.

Impulse - Linking Momentum to Force and Time

Applying a Force

What truly happens when we apply a force?
- Contact begins.
- Maximum compression.
- Contact ends.
Applying a force takes time.

Graph of Force vs. Time

The area under this curve is equal to impulse which equals average force times time.

Formula #1 for Impulse

Formula: $J = F \Delta t$ , where:
- $J$ represents impulse.
- $F$ represents force.
- $\Delta t$ represents the change in time.
Units:
- $\text{kg m/s}^2 \cdot \text{s} = \text{kg m/s}$

Impulse and Momentum

Acceleration causes a change in velocity over time.
Changing velocity means we change momentum.
$F \Delta t$ allows $F$ to actually change velocity.
$F$ provides instantaneous acceleration but no change in velocity since $\Delta v = a \Delta t$ !
$F \Delta t$ also changes momentum!

2 Formulas for Impulse

$J = F_{\text{net, avg}} \Delta t$
$J = \Delta p = m<em>f v</em>f - m<em>i v</em>i$

Combined Formula

$\Delta p = F_{\text{net, avg}} \Delta t$
Directly Proportional:
- Impulse (change in momentum) & average net force
- Impulse (change in momentum) & $\Delta t$

Impulse, Force, and Time

When you kick a ball, is the impulse on your foot vs. the impulse on the ball different?
Are the Forces different?
Is the time of contact different?
No to all three questions.

Impulse and Newton’s Laws

Since force and time are the same for both objects, the impulse from each object must be the same.
Impulse follows Newton’s 3rd Law of Motion.

Impulse: Usefulness

Links force and time to momentum.
Explains concepts of “follow through” and “cushioning.”