Welcome to Dynamics: Momentum and Newton's Laws!

Hey future physicist! This chapter is where mechanics gets really interesting. Instead of just describing motion (kinematics), we now ask: What causes motion? The answer lies in forces, mass, and a powerful concept called momentum. Mastering this section is crucial, as it forms the bedrock for understanding everything from rocket launches to car safety.

Don't worry if these ideas seem complicated; we'll break them down into simple, manageable steps, focusing specifically on the requirements for the Cambridge 9702 syllabus.

1. Mass, Inertia, and Newton's Three Laws (3.1)

1.1 Mass and Inertia

Before discussing forces, we need to understand mass.

  • Definition: Mass is the property of an object that resists change in motion. This resistance is called inertia.
  • The larger the mass, the harder it is to start it moving, stop it moving, or change its direction.

Analogy: Imagine trying to push a parked bicycle versus a massive freight train. The train has vastly more mass, and thus vastly more inertia, resisting any effort to change its state of motion.

1.2 Newton's First Law: The Law of Inertia

Newton's First Law builds directly on the concept of inertia.

  • Statement: An object will remain at rest or continue to move at a constant velocity unless acted upon by a resultant force.
  • Key Point: If the resultant force (\(F_{net}\)) on an object is zero, its acceleration (\(a\)) must be zero. It maintains its current velocity (which might be zero).

1.3 Newton's Second Law: Force and Acceleration

This is perhaps the most famous equation in physics, quantifying how forces cause changes in motion.

Recalling and Applying \(F = ma\)

The resultant force acting on an object is proportional to the rate of change of momentum (we'll look at this later) and is commonly written as:

$$F = ma$$

Where:

  • \(F\) is the resultant force (or net force) acting on the mass (Units: Newtons, N).
  • \(m\) is the mass of the object (Units: kg).
  • \(a\) is the acceleration (Units: m s\(^{-2}\)).

Crucial Understanding: The acceleration (\(a\)) and the resultant force (\(F_{net}\)) are always in the same direction. If you push an object North, it accelerates North (even if it's currently moving South).

1.4 Newton's Third Law: Action and Reaction

Forces always come in pairs.

  • Statement: When object A exerts a force on object B (the 'Action'), object B simultaneously exerts an equal and opposite force on object A (the 'Reaction').

Important Checklist for \(F_{AB} = -F_{BA}\):

  1. The forces are equal in magnitude.
  2. The forces are opposite in direction.
  3. The forces are of the same type (e.g., gravitational, electrical, or contact).
  4. The forces act on different bodies (they never cancel each other out because they are never applied to the same object).

Example: When you jump, you push the Earth down (Action). The Earth pushes you up with an equal and opposite force (Reaction). Since the Earth's mass is huge, its acceleration is negligible.

1.5 Weight (3.1)

Weight is a specific type of force defined by the gravitational pull of a planet.

  • Concept: Weight is the effect of a gravitational field on a mass.
  • Calculation: The weight (\(W\)) of an object is equal to the product of its mass (\(m\)) and the acceleration of free fall (\(g\)).

$$W = mg$$

Quick Review: Mass is the amount of stuff (scalar, measured in kg). Weight is the gravitational force acting on that stuff (vector, measured in N).

Key Takeaway for Newton's Laws

Newton's Laws explain the cause of motion: F1 tells us what happens when forces are balanced; F2 quantifies the acceleration when forces are unbalanced (\(F=ma\)); and F3 ensures that forces always occur in reciprocal pairs acting on different objects.

2. Linear Momentum (3.1, 3.3)

Linear Momentum is a fundamental quantity that combines an object's mass and its velocity. Think of it as the "quantity of motion."

2.1 Definition and Formula

Definition: Linear momentum (\(p\)) is defined as the product of mass (\(m\)) and velocity (\(v\)).

$$p = mv$$

  • Units: The SI unit for momentum is kg m s\(^{-1}\).
  • Vector Quantity: Momentum is a vector. Its direction is the same as the velocity. This is extremely important, especially when dealing with collisions.

Example: A small bullet (low mass) traveling very fast can have the same momentum as a large, slow-moving train carriage (high mass).

Memory Aid: P=MV is easy to remember, but always ensure you use SI units (kg and m s\(^{-1}\)).

2.2 Force as Rate of Change of Momentum (3.1)

Newton's Second Law can be formally defined using momentum.

Definition: Force is defined as the rate of change of momentum.

$$F = \frac{\Delta p}{\Delta t} = \frac{(mv - mu)}{t}$$

(Where \(mu\) is initial momentum and \(mv\) is final momentum.)

Did you know? This definition, \(F = \Delta p / \Delta t\), is more fundamental than \(F=ma\). If mass is constant, we can pull the mass out: \(F = m \frac{\Delta v}{\Delta t}\). Since \(\frac{\Delta v}{\Delta t} = a\), we get back to \(F=ma\). But if mass changes (like a rocket burning fuel), the momentum definition must be used.

Application: Crash Safety

If you want to stop a moving object (change its momentum, \(\Delta p\)), that change must happen over some time, \(\Delta t\).

If \(\Delta t\) is small (hitting a brick wall), the resulting force \(F\) is huge.
If safety features (like airbags, seatbelts, or crumple zones) increase the time \(\Delta t\) needed for the momentum change, the average force \(F\) exerted on the person is reduced.

3. Non-Uniform Motion and Terminal Velocity (3.2)

So far, we mainly discussed motion without considering opposing forces like friction or air resistance. In reality, these resistive forces are crucial.

3.1 Resistive Forces (Friction and Drag)

  • Friction and Viscous/Drag Forces (including air resistance) are forces that always oppose motion.
  • For a simple model, we assume the drag force (air resistance) increases as the speed of the object increases.
  • The syllabus requires a qualitative understanding: we don't need complex equations for drag coefficients, just the idea that resistance grows with speed.

3.2 Motion with Air Resistance in a Gravitational Field

When an object falls in air, the downward force (Weight, \(W\)) is opposed by the upward drag force (Air Resistance, \(D\)).

Step-by-Step: Reaching Terminal Velocity

  1. Start: The object is released. \(v=0\), so Drag \(D=0\). The Resultant Force \(F_{net} = W - D = W\). Acceleration \(a\) is maximum (\(g\)).
  2. Acceleration Phase: As speed \(v\) increases, Drag \(D\) increases. \(F_{net} = W - D\) decreases. Therefore, acceleration \(a\) decreases.
  3. Equilibrium (Terminal Velocity): The object continues to accelerate until the Drag force \(D\) becomes exactly equal to the Weight \(W\). At this point, \(F_{net} = 0\).
  4. Result: Since the resultant force is zero, the acceleration is zero. The object now moves at a constant, maximum velocity called the Terminal Velocity (\(v_T\)).

Analogy: The Skydiver
A skydiver accelerates rapidly until they reach their first terminal velocity (about 50 m/s). When they open their parachute, the surface area increases dramatically, increasing the drag force \(D\). Now \(D > W\), so they decelerate until a much slower, safer terminal velocity is reached for landing.

Key Takeaway for Terminal Velocity

Terminal velocity is reached when the resultant force on an object moving against a resistive force (like air resistance) is zero.

4. Conservation of Momentum and Collisions (3.3)

The principle of conservation of momentum is one of the most powerful concepts in physics. It allows us to predict the outcomes of interactions like collisions and explosions without knowing the exact forces involved.

4.1 Principle of Conservation of Momentum

Statement: For a system of interacting objects, the total linear momentum remains constant, provided there are no external forces acting on the system (a closed system).

In simpler terms, the total momentum before an interaction (collision or explosion) equals the total momentum after the interaction.

$$p_{total, before} = p_{total, after}$$

If we consider two masses, \(m_1\) and \(m_2\), with initial velocities \(u_1\) and \(u_2\), and final velocities \(v_1\) and \(v_2\):

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

Warning: Vector Direction! Since momentum is a vector, you must assign positive and negative directions. If a mass is moving right, its velocity (and momentum) is positive. If it moves left, its velocity (and momentum) is negative. Be consistent!

4.2 Applying Conservation of Momentum (1D and 2D)

One Dimension (1D)

Most problems involve objects moving along a straight line. Use the formula above, carefully substituting positive and negative signs for velocities.

Example (Recoil): A gun (mass \(M\)) fires a bullet (mass \(m\)). Initially, everything is at rest (total momentum = 0). After firing, the total momentum must still be zero:

$$0 = Mv_{gun} + mv_{bullet}$$

This shows that the gun's final velocity \(v_{gun}\) must be in the opposite direction (negative sign) to the bullet's velocity \(v_{bullet}\).

Two Dimensions (2D)

For interactions (like glancing collisions on an air table), momentum is conserved separately in two perpendicular directions (usually x and y axes).

  • Total momentum in the X-direction before collision = Total momentum in the X-direction after collision.
  • Total momentum in the Y-direction before collision = Total momentum in the Y-direction after collision.

Don't worry if this seems tricky at first. It just means breaking every velocity vector into its components before applying the conservation law!

4.3 Types of Interactions: Elastic vs. Inelastic (3.3)

While momentum is always conserved in a closed system (according to the principle), Kinetic Energy (KE) may or may not be conserved during the interaction.

Elastic Collisions
  • Definition: A collision in which both momentum AND total kinetic energy are conserved.
  • No energy is permanently converted into other forms (heat, sound, deformation).
  • Key property: The relative speed of approach before the collision is equal to the relative speed of separation after the collision.

$$Total \, KE_{before} = Total \, KE_{after}$$ $$ \frac{1}{2}m_1 u_1^2 + \frac{1}{2}m_2 u_2^2 = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2 $$

Inelastic Collisions
  • Definition: A collision in which momentum is conserved, but the total kinetic energy is NOT conserved.
  • Kinetic energy is lost (converted into heat, sound, or used to permanently deform the objects).
  • The most extreme inelastic collision is one where the objects stick together (like two clay balls), moving with a common final velocity.

Crucial Reminder: The syllabus emphasizes that some change in kinetic energy may take place, but the momentum of the system is always conserved during interactions.

Summary Table: Collisions
Collision Type Conservation of Momentum Conservation of Kinetic Energy
Elastic Conserved Conserved
Inelastic Conserved NOT Conserved (Lost as heat/sound)
Common Mistake to Avoid

Do NOT assume KE is conserved unless the problem explicitly states the collision is elastic. In most real-world scenarios (car crashes, billiard balls hitting quietly), the collision is inelastic. Always start by conserving momentum.

Key Takeaway for Conservation of Momentum

The conservation of momentum (\(p_{before} = p_{after}\)) is a universal law in a closed system. Use the conservation of KE (\(KE_{before} = KE_{after}\)) ONLY if the collision is elastic.