Welcome to Dynamics!
Welcome to one of the most exciting and fundamental chapters in AS Physics: Dynamics. While kinematics describes how objects move, dynamics explains why they move by looking at the forces responsible for motion.
Whether you are analyzing a car braking, a rocket launching into space, or a skydiver reaching terminal velocity, the principles in this chapter provide the exact rules the universe follows. Don't worry if mechanics has felt tricky in the past; we will break down every concept into small, easy-to-understand steps with plenty of real-world examples.
1. Newton's First Law and the Concept of Inertia
Before diving into calculations, let's explore how objects behave naturally when left alone.
What is Newton's First Law?
Newton's First Law of Motion states that an object will remain at rest or continue to move with a constant velocity in a straight line unless acted upon by a resultant (unbalanced) external force.
In mathematical terms, if the resultant force \(\Sigma F = 0\), then the acceleration \(a = 0\text{ m s}^{-2}\). This means:
• If the object is stationary, it stays stationary.
• If the object is already moving, its speed and direction do not change.
Understanding Inertia and Mass
Inertia is the natural tendency of an object to resist any change in its state of motion. The property of an object that determines its inertia is its mass (\(m\)), measured in kilograms (\(\text{kg}\)).
Analogy: Imagine trying to push a shopping trolley. When it is empty, it is easy to start moving and easy to stop. But when it is fully loaded with heavy groceries, it resists speeding up and takes much more effort to stop. The loaded trolley has greater mass, and therefore greater inertia.
Common Pitfall
Mistake: Thinking that a force is needed to keep an object moving at a constant speed.
Reality: In everyday life, friction and air resistance slow things down, so we have to keep pushing. In the absence of friction (like in deep space), an object will coast forever without any engine running!
Key Takeaway: If forces are balanced (\(F_{\text{net}} = 0\)), there is no acceleration. A change in motion requires a resultant force.
2. Linear Momentum and Newton's Second Law
What is Linear Momentum?
Linear momentum (\(p\)) is defined as the product of an object's mass and its velocity.
\(p = mv\)
• \(p\) = momentum in kilogram metres per second (\(\text{kg m s}^{-1}\)) or Newton seconds (\(\text{N s}\))
• \(m\) = mass in kilograms (\(\text{kg}\))
• \(v\) = velocity in metres per second (\(\text{m s}^{-1}\))
Because velocity is a vector quantity (it has magnitude and direction), momentum is also a vector quantity. Always choose a positive direction (e.g., to the right = \(+\), to the left = \(-\)).
Newton's Second Law of Motion
Formal Definition: The rate of change of momentum of an object is directly proportional to the resultant force acting on it, and takes place in the direction of that force.
Mathematically, we write:
\(F = \frac{\Delta p}{\Delta t}\)
Where \(\Delta p\) is the change in momentum: \(\Delta p = mv - mu\).
Deriving \(F = ma\)
Let's see how the familiar formula \(F = ma\) comes directly from Newton's Second Law when mass remains constant:
1. Start with \(F = \frac{\Delta p}{\Delta t} = \frac{mv - mu}{t}\)
2. Factor out the mass: \(F = \frac{m(v - u)}{t}\)
3. Recall that acceleration is defined as \(a = \frac{v - u}{t}\)
4. Substitute \(a\) into the equation to get: \(F = ma\)
• \(F\) = resultant force in Newtons (\(\text{N}\))
• \(m\) = mass in \(\text{kg}\)
• \(a\) = acceleration in \(\text{m s}^{-2}\)
Definition of the Newton: One Newton is the force required to give a mass of \(1\text{ kg}\) an acceleration of \(1\text{ m s}^{-2}\) in the direction of the force.
Key Takeaway: Resultant force equals the rate of change of momentum. When mass is constant, this simplifies to \(F = ma\).
3. Impulse and Force–Time Graphs
What is Impulse?
Impulse is defined as the product of the average resultant force acting on an object and the time for which it acts. Rearranging Newton's Second Law gives:
\(\text{Impulse} = F\Delta t = \Delta p = mv - mu\)
• Impulse is measured in Newton seconds (\(\text{N s}\)) or \(\text{kg m s}^{-1}\).
• Impulse equals the change in momentum.
Force–Time Graphs
In real life, forces during impacts are rarely constant. On a graph of Force (vertical axis) against Time (horizontal axis):
Area under a Force–Time graph = Impulse = Change in Momentum (\(\Delta p\))
• For a rectangle: \(\text{Area} = \text{Force} \times \text{time}\)
• For a triangle: \(\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}\)
Real-World Application: Impact Safety
To bring a moving person to rest, their momentum must change by a fixed amount (\(\Delta p = \text{constant}\)). Since \(\Delta p = F\Delta t\):
• If we increase the time of impact (\(\Delta t\)), the impact force (\(F\)) is significantly reduced.
• This is the scientific principle behind car crumple zones, airbags, seatbelts, cushioned running shoes, and bending your knees when you land from a jump.
Key Takeaway: Impulse is the change in momentum and corresponds to the area under a force–time graph. Increasing contact time decreases the destructive peak force.
4. Newton's Third Law of Motion
The Law of Interaction
Newton's Third Law states: When body A exerts a force on body B, body B exerts an equal in magnitude and opposite in direction force of the same type on body A.
Four Essential Conditions for a Newton-3 Pair
For two forces to form a genuine Newton's Third Law pair, they MUST:
1. Be equal in magnitude.
2. Act in exactly opposite directions.
3. Be of the exact same type (e.g., both gravitational, both normal contact, both electrostatic).
4. Act on two different bodies.
Common Misunderstanding: A Book Resting on a Table
Consider a book resting on a flat table:
• The Earth pulls downward on the book with gravity (Weight of the book).
• The table pushes upward on the book with an equal normal contact force (\(R\)).
Are these two forces a Newton-3 pair? No! Although they are equal and opposite, they act on the same object (the book) and are of different types (gravitational vs. electrostatic/contact). They are an example of Newton's First Law equilibrium.
The actual Newton-3 pairs are:
• Pair 1: Earth pulls book down (gravity) \(\leftrightarrow\) Book pulls Earth up (gravity).
• Pair 2: Book pushes table down (contact) \(\leftrightarrow\) Table pushes book up (contact).
Key Takeaway: Newton-3 pairs always act on different bodies and represent the exact same type of fundamental force.
5. Principle of Conservation of Linear Momentum
The Conservation Principle
The Principle of Conservation of Linear Momentum states that in any closed system with no external resultant forces, the total linear momentum before an event equals the total linear momentum after the event.
\(\text{Total Momentum Before} = \text{Total Momentum After}\)
\(m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2\)
Step-by-Step Problem Solving Strategy
1. Pick a positive direction (e.g., right = \(+\), left = \(-\)).
2. Write down the mass and initial velocity of each object, assigning a minus sign to any velocity pointing left.
3. Write down expressions for initial total momentum and final total momentum.
4. Set them equal: \(\Sigma p_{\text{initial}} = \Sigma p_{\text{final}}\).
5. Solve for the unknown variable.
Elastic vs. Inelastic Collisions
In all collisions in a closed system, momentum is always conserved and total energy is always conserved. However, kinetic energy behaves differently:
• Elastic Collision: Kinetic energy is conserved (\(\text{Total } E_k \text{ before} = \text{Total } E_k \text{ after}\)). Particles bounce apart with no loss of mechanical energy.
• Inelastic Collision: Kinetic energy is NOT conserved. Some \(E_k\) is converted into internal energy, heat, or sound (\(\text{Total } E_k \text{ after} < \text{Total } E_k \text{ before}\)). If objects stick together after collision, it is perfectly inelastic.
• Explosions: Kinetic energy increases (\(\text{Total } E_k \text{ after} > \text{Total } E_k \text{ before}\)). Total initial momentum is often zero (e.g., a stationary cannon firing a cannonball: \(0 = m_{\text{cannon}}v_{\text{cannon}} + m_{\text{ball}}v_{\text{ball}}\)).
Key Takeaway: Momentum is always conserved in closed systems. Check kinetic energy before and after to determine whether a collision is elastic or inelastic.
6. Resistive Forces and Terminal Velocity
Drag and Air Resistance
When an object moves through a fluid (liquid or gas), it experiences a resistive force known as drag or air resistance (\(D\)).
Key properties of drag force:
• It always acts in the direction opposite to motion.
• Its magnitude increases rapidly as the speed of the object increases (\(D \propto v^2\) at high speeds).
• It depends on the cross-sectional surface area and shape of the object.
The Journey to Terminal Velocity (Step-by-Step)
Let's follow a skydiver jumping out of an aircraft:
Stage 1: Immediately after jumping (\(v = 0\))
• Air resistance is zero (\(D = 0\)).
• The only force acting is the downward weight (\(W = mg\)).
• Resultant force: \(F_{\text{net}} = W\).
• Initial acceleration is at its maximum: \(a = g \approx 9.81\text{ m s}^{-2}\).
Stage 2: As speed increases (\(v\) rises)
• Air resistance increases as speed increases.
• Resultant downward force decreases: \(F_{\text{net}} = W - D\).
• Acceleration decreases (\(a < g\)), but the skydiver is still speeding up (rate of increase is slowing down).
Stage 3: Terminal Velocity reached
• Drag grows until it matches the skydiver's weight: \(D = W\).
• Resultant force is now zero: \(F_{\text{net}} = 0\).
• Acceleration becomes zero: \(a = 0\text{ m s}^{-2}\).
• The skydiver falls at a constant maximum velocity called terminal velocity (\(v_t\)).
Stage 4: Parachute opens
• The massive surface area causes drag to suddenly become much larger than weight (\(D > W\)).
• Resultant force acts upward, causing rapid deceleration (speed decreases).
• As speed drops, drag decreases until \(D = W\) again.
• A new, much lower terminal velocity is established, allowing a safe landing.
Key Takeaway: Terminal velocity occurs when the upward resistive force equals the downward gravitational force, creating zero resultant force and zero acceleration.
7. Connected Particles and Apparent Weight
Objects in Lifts (Apparent Weight)
When standing on a weighing scale in a lift, the scale reads the normal contact force (\(R\)), which is your apparent weight, not necessarily your true gravitational weight (\(W = mg\)).
• Lift stationary or moving at constant velocity:
\(a = 0 \implies R - mg = 0 \implies R = mg\) (Scale reads normal weight).
• Lift accelerating upwards (or decelerating downwards):
\(F_{\text{net}} = R - mg = ma \implies R = m(g + a)\) (You feel heavier; scale reading increases).
• Lift accelerating downwards (or decelerating upwards):
\(F_{\text{net}} = mg - R = ma \implies R = m(g - a)\) (You feel lighter; scale reading decreases).
• Free fall (cable snaps, \(a = g\)):
\(R = m(g - g) = 0\text{ N}\) (Weightlessness; you float above the scale).
Quick Chapter Summary
• Newton's 1st Law: Balanced forces mean zero acceleration (\(F_{\text{net}} = 0 \implies v = \text{constant}\)).
• Newton's 2nd Law: \(F = \frac{\Delta p}{\Delta t}\) and \(F = ma\) (for constant mass).
• Newton's 3rd Law: Forces occur in equal and opposite pairs of the same type on different bodies.
• Momentum: \(p = mv\), a vector quantity conserved in all isolated interactions.
• Impulse: \(F\Delta t = \Delta p = \text{Area under } F\text{-}t \text{ graph}\).
• Terminal Velocity: Occurs when drag equals weight, leading to \(a = 0\text{ m s}^{-2}\).