Momentum (1D and implied 2D)

Physics Study Notes: Momentum (1D and Implied 2D)

Hello! Welcome to the world of momentum. Ever wondered how a tiny, fast-moving bullet can have a huge impact, or how airbags in cars save lives? The answer lies in the concept of momentum. In these notes, we'll break down everything you need to know about momentum, impulse, collisions, force–time graphs, and 2D vector changes. Let's get moving!

1. What is Momentum? It's Mass in Motion!

Imagine a bowling ball and a tennis ball rolling towards you at the same, slow speed. Which one would you rather stop? The tennis ball, of course! Why? Because the bowling ball has more mass. Now, imagine two tennis balls. One is thrown gently, and the other is fired from a cannon. Which is harder to stop? The one from the cannon, because it has a much higher velocity.

This idea of how 'hard an object is to stop' is what physicists call linear momentum. It’s a measure of an object's 'quantity of motion', combining both its mass and its velocity.

Defining Linear Momentum (p)

Momentum is the product of an object's mass and its velocity. We use the letter p to represent momentum.

The formula is simple:

\( p = mv \)

Where:
p = momentum (in \(\text{kg m s}^{-1}\) or \(\text{N s}\))
m = mass (in \(\text{kg}\))
v = velocity (in \(\text{m s}^{-1}\))

Momentum is a Vector!

This is SUPER important! Because velocity has a direction, momentum also has a direction. In 1D problems, we show this with positive (+) and negative (-) signs.

For example, if we decide 'right' is the positive direction:
- A car moving to the right has a positive (+) momentum.
- A car moving to the left has a negative (-) momentum.

Common Mistake Alert: Forgetting to include the correct sign for velocity is the #1 reason students make mistakes in momentum problems. Always define your positive direction first!

Units of Momentum

The standard unit is kilogram-metres per second (\(\text{kg m s}^{-1}\)). You might also see it written as newton-seconds (\(\text{N s}\)). They are equivalent: \(1\text{ kg m s}^{-1} = 1\text{ N s}\).

Quick Review
What is Momentum? A measure of an object's mass in motion.
Formula: \(p = mv\)
Key Property: It's a vector! Direction matters.

Key Takeaway: Momentum tells you how much 'oomph' a moving object has. More mass or more velocity means more momentum.

2. Change in Momentum & Impulse

To change an object's velocity (to accelerate it), you need to apply a net force. Since momentum depends on velocity, a net force is needed to change an object's momentum. This is the core of Newton's Second Law!

Newton's Second Law (The Momentum Version)

You probably know Newton's Second Law as \(F = ma\). But there's a more fundamental way to state it:

The net force acting on an object is equal to the rate of change of its momentum.

In formula terms:

\( F_{\text{net}} = \frac{\Delta p}{\Delta t} = \frac{p_{\text{final}} - p_{\text{initial}}}{\Delta t} = \frac{mv - mu}{\Delta t} \)

Where:
\(F_{\text{net}}\): Net force (\(\text{N}\))
\(\Delta p\): Change in momentum (\(\text{kg m s}^{-1}\))
\(\Delta t\): Time interval over which the force acts (\(\text{s}\))

Introducing Impulse

Rearranging the formula gives:

\( \text{Impulse} = F_{\text{net}} \Delta t = \Delta p = mv - mu \)

Impulse is defined as the product of the net force and the time duration of impact, and it is numerically equal to the change in momentum.

Force–Time (\(F\)–\(t\)) Graphs

In most real-life collisions, the force is not constant; it increases to a peak and drops back to zero. On a force–time (\(F\)–\(t\)) graph:

• Area under the \(F\)–\(t\) graph = Impulse = Change in momentum (\(\Delta p\)).
• If force is non-constant, the average net force is \(F_{\text{avg}} = \frac{\text{Area}}{\Delta t} = \frac{\Delta p}{\Delta t}\).

Real World Applications:
- Airbags & Crumple Zones: When a car crashes, the change in momentum (\(\Delta p\)) is fixed. Airbags and crumple zones increase the time (\(\Delta t\)) of impact. Since \(F_{\text{avg}} = \frac{\Delta p}{\Delta t}\), increasing \(\Delta t\) significantly reduces the average impact force on passengers.
- Bending knees when landing: Increases the stopping time, reducing the impact force on your joints.
- Follow-through in sports: Extends contact time (\(\Delta t\)) between the racket/club and the ball, delivering a greater impulse (\(F\Delta t\)) and yielding a higher exit velocity (\(v\)).

3. The Law of Conservation of Momentum

The Law of Conservation of Linear Momentum states that in a closed system (where no external net force acts), the total linear momentum before an interaction equals the total linear momentum after.

For a collision between two objects (1 and 2):

\( m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2 \)

Relation to Newton's Third Law

• During contact, force on 1 by 2 (\(F_{12}\)) and force on 2 by 1 (\(F_{21}\)) form an action-reaction pair: \(F_{12} = -F_{21}\).
• Contact time \(\Delta t\) is identical for both.
• Therefore, the impulse on 1 equals the negative impulse on 2: \(F_{12}\Delta t = -F_{21}\Delta t \implies \Delta p_1 = -\Delta p_2\).
• Rearranging gives \(\Delta p_1 + \Delta p_2 = 0\), meaning total system momentum does not change.

Experimental Verification

In the laboratory, conservation of momentum is commonly verified using:

• Friction-compensated inclined runway: The track is tilted slightly so the component of gravity down the slope exactly balances friction.
• Linear Air Track: Air cushion minimizes friction between gliders and track.
• Light gates and data loggers: Card interrupters of known width pass through light gates to measure initial velocities (\(u_1, u_2\)) and final velocities (\(v_1, v_2\)). Calculating \(m_1 u_1 + m_2 u_2\) and comparing it with \(m_1 v_1 + m_2 v_2\) verifies momentum conservation.

4. Types of Collisions

While total linear momentum is always conserved in a closed system, kinetic energy (\(E_k = \frac{1}{2}mv^2\)) may or may not be conserved.

Elastic Collisions

- Momentum is conserved.
- Kinetic energy is conserved: \(\frac{1}{2}m_1 u_1^2 + \frac{1}{2}m_2 u_2^2 = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2\).
No kinetic energy is transformed into internal energy, heat, or sound.

Inelastic Collisions

- Momentum is conserved.
- Kinetic energy is NOT conserved (it decreases): Some \(E_k\) is transformed into internal energy (deformation), heat, or sound.

Perfectly Inelastic Collisions

- The objects stick together and move with a single common final velocity \(v_{\text{final}}\).
- Maximum loss of kinetic energy.
- Equation: \( m_1 u_1 + m_2 u_2 = (m_1 + m_2) v_{\text{final}} \)

5. Momentum in 2D (Vector Nature)

Momentum is a vector quantity in two dimensions as well. When resolving 2D interactions (e.g. a ball bouncing off a smooth vertical wall at an angle \(\theta\)):

• Resolve momentum into perpendicular components: \(x\)-component (normal to the wall) and \(y\)-component (parallel to the wall).
• Calculate change in momentum component by component: \(\Delta p_x = m(v_x - u_x)\) and \(\Delta p_y = m(v_y - u_y)\).
• For a smooth wall, no force acts parallel to the wall (\(\Delta p_y = 0\)), while the normal force creates impulse \(\Delta p_x = F_N \Delta t\).
• Vector Subtraction / Addition: Overall impulse is \(\vec{J} = \Delta \vec{p} = \vec{p}_{\text{final}} - \vec{p}_{\text{initial}}\).

6. Worked Example: 1D Collision

A 3 kg cart (Cart A) moving right at \(2\text{ m s}^{-1}\) collides with a 1 kg cart (Cart B) at rest. They stick together after colliding. Find their final velocity.

Step 1: Define direction
Let Right be positive (+).

Step 2: List variables
\(m_A = 3\text{ kg}\), \(u_A = +2\text{ m s}^{-1}\)
\(m_B = 1\text{ kg}\), \(u_B = 0\text{ m s}^{-1}\)

Step 3: Apply conservation of momentum
\( m_A u_A + m_B u_B = (m_A + m_B) v_{\text{final}} \)
\( (3)(+2) + (1)(0) = (3 + 1) v_{\text{final}} \)
\( 6 = 4 v_{\text{final}} \)
\( v_{\text{final}} = +1.5\text{ m s}^{-1} \)

Conclusion: The carts move together to the right at \(1.5\text{ m s}^{-1}\).