Physics (9630) | Momentum

Momentum: The Physics of Motion and Impact (Syllabus 3.2.6)

Welcome to the chapter on Momentum! This concept is fundamental to understanding how objects interact, collide, and explode. It connects the ideas of mass and velocity and forms the backbone of mechanics, allowing us to predict the outcomes of dramatic events—from a tiny snooker ball collision to a massive rocket launch.

Don't worry if you sometimes confuse momentum with kinetic energy; while they are related, they describe different aspects of motion. We'll break down the core definitions, the powerful conservation laws, and look at how these principles keep you safe in a car accident!

1. Defining Momentum (\(p\))

In simple terms, momentum describes how much "oomph" an object has. It's a measure of how hard it is to stop a moving object.

Key Definition and Formula

Momentum (\(p\)) is defined as the product of an object's mass (\(m\)) and its velocity (\(v\)).

• Formula: \(p = mv\)
• Units: Since mass is measured in kilograms (kg) and velocity in metres per second (m s\(^{-1}\)), the unit for momentum is kg m s\(^{-1}\).
• Vector Quantity: Momentum is a vector quantity. This means it has both magnitude (size) and direction. When solving problems, the direction (e.g., left or right, positive or negative) is absolutely critical!

Analogy: Why direction matters

Imagine two cars, both having the same mass and speed (say, 50 km/h).

Car A is moving at 50 km/h east.
Car B is moving at 50 km/h west.

They have the same magnitude of momentum, but their total vector momentum is zero if they crash head-on because the directions cancel out. You must always assign a positive direction (e.g., "to the right is positive") before starting a calculation.

2. Force, Impulse, and Change in Momentum

Momentum isn't just about objects moving; it's also about how forces affect that motion.

Newton's Second Law (Revisited)

We already know Newton’s Second Law as \(F = ma\). However, Newton originally stated the law in terms of momentum:

"The resultant force acting on an object is equal to the rate of change of momentum."

Formula for Force: \[F = \frac{\Delta (mv)}{\Delta t} = \frac{\Delta p}{\Delta t}\]

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

Impulse: The Measure of Impact

When a force \(F\) acts on an object for a time \(\Delta t\), the product \(F\Delta t\) is called the Impulse.

The syllabus defines Impulse as the change in momentum (\(\Delta p\)).

• Impulse-Momentum Theorem: \[\text{Impulse} = F\Delta t = \Delta (mv)\] Where \(F\) is the constant force applied during the time interval \(\Delta t\).
• Units of Impulse: The unit of impulse is the Newton second (N s). Since impulse equals the change in momentum, its unit must also be kg m s\(^{-1}\) (1 N s = 1 kg m s\(^{-1}\)).

The Power of the Impulse Equation

The relationship \(F\Delta t = \Delta p\) is extremely important, especially in safety engineering.

If an object's change in momentum (\(\Delta p\)) is fixed (e.g., a car coming to a complete stop, \(\Delta p = 0 - mv\)), you can see that force and time are inversely related:

\[F \propto \frac{1}{\Delta t}\]

• To reduce the impact force (\(F\)) felt during a stop, you must increase the contact time (\(\Delta t\)).

Example: When catching a cricket ball, you move your hands backwards. This increases the time (\(\Delta t\)) over which the ball's momentum changes, reducing the force (\(F\)) on your hands and making the catch less painful!

3. Force-Time Graphs

What if the force isn't constant? In most real-world impacts (like a football being kicked), the force rises sharply and then drops. This is where graphs come in handy.

The syllabus requires you to know the significance of the area under a force-time graph.

• Significance: The area under a force-time graph gives the Impulse, which is equal to the change in momentum (\(\Delta p\)).

For quantitative questions involving variable force, you will calculate the area of the shape shown (usually a rectangle or a triangle) to find the change in momentum.

4. The Conservation of Linear Momentum

This is one of the most powerful laws in Physics. It governs all interactions, from collisions to explosions, provided the system is closed.

The Principle

The Principle of Conservation of Linear Momentum states that for a closed system (where no external forces act, like friction or gravity), the total momentum before an interaction is equal to the total momentum after the interaction.

Initial Total Momentum = Final Total Momentum

Mathematical Form (for two bodies in 1D): \[m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2\]

Where:

• \(m_1\), \(m_2\) are the masses of the objects.
• \(u_1\), \(u_2\) are the initial velocities (before interaction).
• \(v_1\), \(v_2\) are the final velocities (after interaction).

Applying the Principle (1D Problems Only)

The syllabus restricts quantitative problems to motion in one dimension (1D). This simplifies things greatly!

Step-by-Step for Momentum Problems:

1. Define Direction: Assign a positive direction (e.g., right = positive). All velocities in the opposite direction must be substituted as negative values.
2. List Variables: Write down \(m_1, u_1, m_2, u_2\), etc., including signs.
3. Apply Conservation: Set up the equation \(m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2\).
4. Solve: Solve for the unknown velocity (the final sign of the answer tells you the final direction).

Common Mistake to Avoid: Forgetting that velocity is a vector! A trolley moving left at 2 m s\(^{-1}\) must be entered as \(v = -2\) m s\(^{-1}\).

5. Types of Interactions (Collisions and Explosions)

While momentum is always conserved in a closed system, kinetic energy may or may not be conserved. This distinction defines the type of interaction.

1. Elastic Collisions

• Definition: Both momentum AND kinetic energy (KE) are conserved.
• KE Check: Total initial KE = Total final KE. \[\frac{1}{2}m_1 u_1^2 + \frac{1}{2}m_2 u_2^2 = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2\]
• Example: Collisions between gas molecules or highly elastic rubber balls.

2. Inelastic Collisions

• Definition: Momentum is conserved, but kinetic energy is NOT conserved.
• KE Change: Some initial KE is converted into other forms, like heat, sound, or permanent deformation energy (e.g., crumpling). Total final KE is less than total initial KE.
• Perfectly Inelastic Collision: The objects stick together after the collision and move with a common final velocity (\(v_1 = v_2 = v\)). Example: A train carriage coupling with another.

3. Explosions (Reverse Collisions)

• Definition: The total initial momentum is usually zero (if the object starts at rest). The objects move apart.
• KE Change: Chemical or potential energy (stored within the system) is converted into KE, so the total KE increases.
• Momentum Conservation: If an object of mass \(M\) initially at rest splits into two pieces, \(m_1\) and \(m_2\), the total momentum remains zero. The two pieces must move in opposite directions to cancel their momentum vectors. \[0 = m_1 v_1 + m_2 v_2\] \[m_1 v_1 = -m_2 v_2\] Example: Firing a cannon or a rifle (recoil).

6. Relationship between Impact Forces and Contact Times

This is the practical application of the Impulse-Momentum Theorem (\(F\Delta t = \Delta p\)). In safety design, we are always trying to minimize damage, which means minimizing the force felt during an impact.

Since the change in momentum (\(\Delta p\)) during an impact is often fixed by the initial conditions (how fast the car was going), the only variable we can manipulate to save lives is the time of impact (\(\Delta t\)).

Real-World Safety Features

• Crumple Zones (Cars): These are designed to deform (crumple) in a collision. By crushing, they extend the duration of the collision (\(\Delta t\)) from milliseconds to slightly longer milliseconds, drastically reducing the force (\(F\)) transmitted to the driver and passengers.
• Packaging Materials (e.g., Foam): Fragile items are wrapped in soft, easily compressed materials. If the package is dropped, the foam squashes, increasing the time (\(\Delta t\)) it takes for the item to stop, thus reducing the force (\(F\)) exerted on it.
• Airbags: They inflate and cushion the driver/passenger. They spread the force over a larger area and, crucially, increase the time (\(\Delta t\)) it takes for the head/body to decelerate to zero velocity.

This concept shows how simple physics equations have profound, life-saving consequences when applied practically!