Physics (9702) | Linear momentum and its conservation

Welcome to the Chapter on Linear Momentum!

Hello! This chapter is fundamental to understanding how things move and interact. If you've ever played pool, watched a car crash, or seen a rocket launch, you've witnessed momentum in action.
Don't worry if concepts like "conservation" seem abstract at first. We will break them down using clear analogies and practical steps.

In this section, you will learn the definition of momentum, how it relates to force (Newton's Laws), and the crucial principle that governs all interactions in the universe: the Conservation of Momentum.

1. Defining Linear Momentum (p)

1.1 What is Momentum?

Momentum is essentially the measure of how difficult it is to stop a moving object. We often call it the "quantity of motion" an object has.

The official definition of linear momentum is:

Linear momentum, \(p\), is the product of an object's mass and its velocity.

1.2 The Momentum Formula

The formula is simple and must be memorised:

$$p = mv$$

Where:
$p$ is the linear momentum (${\text{kg}\text{ m s}^{-1}}$ or ${\text{N s}}$)
$m$ is the mass (${\text{kg}}$)
$v$ is the velocity (${\text{m s}^{-1}}$)

Key Point: Momentum is a Vector!

This is extremely important! Since velocity ($v$) is a vector (it has magnitude and direction), momentum ($p$) must also be a vector quantity.

When solving problems, you must always assign a direction (e.g., positive for right, negative for left) and stick to it consistently.

Key Takeaway for Section 1: Momentum is mass times velocity ($p=mv$), and its direction is the same as the velocity.

2. Force and Rate of Change of Momentum

You learned about Newton’s Second Law ($F=ma$) in Dynamics. This law has a more fundamental form that uses momentum, which is required by the syllabus.

2.1 Defining Force

Newton's Second Law of Motion can be formally stated in terms of momentum:

Force is defined as the rate of change of linear momentum.

In mathematical terms, the resultant force (\(F\)) acting on an object is:

$$F = \frac{\Delta p}{\Delta t}$$

Where \(\Delta p\) is the change in momentum and \(\Delta t\) is the time taken for that change.

2.2 Linking F = ma

Don't worry, $F=ma$ is still valid! We can easily show how this familiar equation comes from the momentum definition, assuming the mass ($m$) is constant:

1. \(F = \frac{\Delta p}{\Delta t}\)
2. Since \(p = mv\), the change in momentum is \(\Delta p = m \Delta v\). (Mass is constant)
3. \(F = \frac{m \Delta v}{\Delta t}\)
4. Since acceleration \(a = \frac{\Delta v}{\Delta t}\), we get:

$$F = ma$$

The key insight here is that force causes momentum to change. If you apply a large force over a short time (like kicking a ball), you cause a large rate of change of momentum.

Key Takeaway for Section 2: Force is fundamentally the rate at which momentum changes ($F = \Delta p / \Delta t$).

3. The Principle of Conservation of Momentum

3.1 The Principle Defined (3.3, L.O. 1)

This principle is one of the most powerful rules in Physics, especially when dealing with collisions and explosions.

Principle of Conservation of Linear Momentum (PCLM):
For a system of interacting objects, the total linear momentum remains constant, provided no net external force acts on the system.

What does "No Net External Force" mean?

An external force is a force originating outside the system (e.g., friction, gravity, air resistance).
Internal forces are the forces acting between the objects within the system (e.g., the force of object A pushing object B during a collision).

During a collision or explosion, the internal forces are typically huge compared to external forces like friction or gravity. For calculations, we usually assume the system is isolated, meaning the net external force is zero, and PCLM applies perfectly.

Analogy: Imagine two bumper cars colliding. The huge push between the cars (internal force) conserves momentum. We ignore the tiny friction with the ground (external force) for the short time of impact.

3.2 Applying the Principle (3.3, L.O. 2)

Mathematically, the principle states:

$$ \text{Total Momentum Before Interaction} = \text{Total Momentum After Interaction} $$

For two objects (1 and 2):

$$m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$$

Where $u$ is initial velocity and $v$ is final velocity. Remember to treat $u$ and $v$ as vectors (use positive and negative signs to indicate direction!).

Example: Recoil (Explosion)

When a cannon fires a shell, the total momentum of the cannon + shell system *before* firing is zero (since both are stationary). Since momentum must be conserved:

Momentum before (0) = Momentum after

$$0 = (m_{\text{cannon}} v_{\text{cannon}}) + (m_{\text{shell}} v_{\text{shell}})$$

This shows that the cannon must recoil (move backwards) with a momentum equal in magnitude but opposite in direction to the shell's momentum.

Memory Aid: PCLM only works when the system is isolated (no police/friction/gravity stopping it!). Total momentum is a constant.

4. Elastic and Inelastic Interactions

Momentum is always conserved in a collision (as long as external forces are negligible). However, Kinetic Energy (KE) may or may not be conserved, which leads to two types of interactions (3.3, L.O. 4).

Recall the KE formula: \(E_k = \frac{1}{2} mv^2\).

4.1 Elastic Collisions (3.3, L.O. 3)

An elastic collision is an interaction where:
1. Momentum is conserved.
2. Total Kinetic Energy is conserved.

$$ \text{Total KE}_{\text{initial}} = \text{Total KE}_{\text{final}} $$

$$ \frac{1}{2} m_1 u_1^2 + \frac{1}{2} m_2 u_2^2 = \frac{1}{2} m_1 v_1^2 + \frac{1}{2} m_2 v_2^2 $$

Relative Speed (A Special Feature of Elastic Collisions)

In an elastic collision, there is another key relationship you must recall (3.3, L.O. 3):

The relative speed of approach is equal to the relative speed of separation.

$$ u_1 - u_2 = v_2 - v_1 $$

(Note: $u_1$ and $u_2$ are initial velocities, $v_1$ and $v_2$ are final velocities. If one object is moving toward the other, you must be careful with vector signs! The formula above is often written using speeds, assuming the objects are moving towards each other initially.)
For example, if object A approaches object B at 5 $m/s$, they must separate at 5 $m/s$.

4.2 Inelastic Collisions (3.3, L.O. 4)

An inelastic collision is an interaction where:
1. Momentum is conserved. (Always!)
2. Total Kinetic Energy is NOT conserved.

In an inelastic collision, some initial KE is lost, usually converted into other forms like:

* Heat energy (due to friction or impact)
* Sound energy
* Potential energy of deformation (crumpling a car bumper)

Therefore:

$$ \text{Total KE}_{\text{initial}} > \text{Total KE}_{\text{final}} $$

Perfectly Inelastic Collisions

A special case is a perfectly inelastic collision, where the maximum amount of KE is lost consistent with momentum conservation. This happens when the two objects stick together and move as one combined mass with a single final velocity ($v_1 = v_2 = V$).
Example: A bullet embedding itself in a block of wood.

Key Takeaway for Section 4: Momentum is always conserved. Energy distinguishes collisions: KE is conserved (Elastic) or KE is lost (Inelastic).

5. Solving Momentum Problems in 1D and 2D

Applying the conservation principle involves careful use of vector addition.

5.1 One-Dimensional (1D) Problems

When objects move along a straight line, vector directions are simple: use positive (+) and negative (-) signs.

Step-by-Step Guide for 1D:

1. Define Directions: Choose one direction (e.g., right) as positive. The opposite direction (left) is negative.
2. Calculate Initial Momentum: Find \(P_{\text{initial}} = m_1 u_1 + m_2 u_2\). (Include signs for $u_1$ and $u_2$).
3. Calculate Final Momentum: Find \(P_{\text{final}} = m_1 v_1 + m_2 v_2\). (Use unknown variables for $v$'s whose direction is not yet known).
4. Apply PCLM: Set \(P_{\text{initial}} = P_{\text{final}}\) and solve for the unknown velocity.

Example: A 2 kg ball moving right at 4 m/s (+8 kg m/s) hits a 1 kg ball moving left at 3 m/s (-3 kg m/s).
Initial Total Momentum = \(+8 + (-3) = +5\text{ kg}\text{ m s}^{-1}\).
The final total momentum must also be \(+5\text{ kg}\text{ m s}^{-1}\).

5.2 Two-Dimensional (2D) Problems (3.3, L.O. 2)

For interactions (like glancing collisions on a pool table) where objects move at angles, momentum is still conserved, but we must treat the vectors component by component.

Because momentum is a vector, its conservation applies independently along any set of perpendicular axes (usually the x-axis and y-axis).

Rule for 2D Conservation:

1. X-direction: \(\sum p_{x, \text{initial}} = \sum p_{x, \text{final}}\)
2. Y-direction: \(\sum p_{y, \text{initial}} = \sum p_{y, \text{final}}\)

Step-by-Step Guide for 2D:

1. Resolve Initial Velocities: Break down all initial velocities ($u$) into $u_x$ and $u_y$ components.
2. Apply PCLM in X: Set the sum of initial x-momenta equal to the sum of final x-momenta.
3. Apply PCLM in Y: Set the sum of initial y-momenta equal to the sum of final y-momenta.
4. Solve: You will now have two simultaneous equations to solve for the unknown velocity components ($v_x$ and $v_y$).

Don't worry if this seems tricky at first! Remember that a 2D problem is just two separate 1D problems stacked together.

Did you know? This principle is why rockets work! A rocket throws mass (hot exhaust gases) downward with high momentum, causing the rocket to gain an equal momentum upward (recoil). Momentum is conserved for the rocket + exhaust system.

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Chapter Summary: Linear Momentum and Conservation

Momentum Basics

• Momentum \(p = mv\) is a vector quantity.
• Force is the rate of change of momentum: \(F = \frac{\Delta p}{\Delta t}\).

Conservation of Momentum (PCLM)

• Statement: The total momentum of an isolated system (no net external force) is conserved.
• Formula: \(m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2\) (treating velocities as vectors).
• For 2D, momentum must be conserved separately along perpendicular axes.

Energy Conservation in Interactions

• Elastic Collision: Momentum and total Kinetic Energy are conserved. Relative speed of approach = relative speed of separation.
• Inelastic Collision: Momentum is conserved, but total Kinetic Energy is not conserved (it is lost, e.g., to heat or deformation).