State conservation of momentum is a fundamental principle in physics that asserts the total momentum of a closed system remains constant over time, provided no external forces act upon it. This concept is pivotal in the study of dynamics, particularly within the 'Linear Momentum and its Conservation' chapter for AS & A Level Physics - 9702. Understanding momentum conservation aids in analyzing collisions, explosions, and various motion scenarios, thereby forming a cornerstone for both theoretical and applied physics.

Key Concepts

Definition of Momentum

Momentum, often denoted by **p**, is a vector quantity defined as the product of an object's mass (**m**) and its velocity (**v**): $$p = m \cdot v$$ Momentum quantifies the motion of an object and its tendency to continue moving. It is crucial in understanding how objects interact, especially during collisions and interactions.

Law of Conservation of Momentum

The law of conservation of momentum states that within a closed system (where no external forces are acting), the total momentum before any interaction is equal to the total momentum after the interaction. Mathematically, for two interacting objects: $$m_1 \cdot u_1 + m_2 \cdot u_2 = m_1 \cdot v_1 + m_2 \cdot v_2$$ where:

m₁ and m₂ are the masses of the objects.
u₁ and u₂ are the initial velocities.
v₁ and v₂ are the final velocities.

This principle is fundamental in analyzing collisions, whether elastic or inelastic.

Types of Collisions

Momentum conservation applies to all types of collisions, but the nature of the collision determines additional characteristics:

Elastic Collisions: Both momentum and kinetic energy are conserved. Objects bounce off each other without permanent deformation.
Inelastic Collisions: Momentum is conserved, but kinetic energy is not. Objects may stick together, resulting in deformation.
Perfectly Inelastic Collisions: A special case of inelastic collisions where objects stick together post-collision, maximizing the loss of kinetic energy.

Impulse and Momentum Change

Impulse is the change in momentum of an object when a force is applied over a time interval. It is given by: $$Impulse (J) = F \cdot \Delta t = \Delta p$$ where:

F is the force applied.
Δt is the time duration.
Δp is the change in momentum.

This relationship is crucial in understanding how forces affect motion over time, particularly in scenarios involving collisions and impacts.

Center of Mass

The center of mass is the point in a system where the mass can be considered to be concentrated for analyzing motion. In a closed system, the center of mass moves with a constant velocity if no external forces act upon it. This concept simplifies the application of momentum conservation, especially in multi-body systems.

Relative Momentum

Momentum is relative to the frame of reference. Changing the observer's frame can alter the perceived momentum of objects. However, the conservation principle holds true across all inertial frames, ensuring that total momentum remains constant regardless of the observer's state of motion.

Mathematical Derivation of Conservation of Momentum

To derive the conservation of momentum, consider Newton's Second and Third Laws:

Newton's Second Law: The rate of change of momentum is equal to the applied force. $$F = \frac{dp}{dt}$$
Newton's Third Law: For every action, there is an equal and opposite reaction. $$F_{12} = -F_{21}$$

For a closed system, the internal forces cancel out: $$\frac{d}{dt}(p_1 + p_2 + \dots + p_n) = \sum F_{\text{external}} = 0$$ Thus, the total momentum remains constant: $$p_{\text{total, initial}} = p_{\text{total, final}}$$

Applications of Momentum Conservation

Momentum conservation is applied in various real-world scenarios, including:

Automotive Safety: Designing airbags and crumple zones to manage momentum changes during collisions.
Aerospace Engineering: Calculating the momentum of rockets and space vehicles for propulsion.
Sports: Analyzing impacts in games like billiards, football, and tennis.
Astronomy: Understanding the dynamics of celestial bodies during interactions and collisions.

Conservation in Multi-Particle Systems

In systems with multiple particles, the total momentum is the vector sum of individual momenta: $$\mathbf{p}_{\text{total}} = \sum_{i=1}^{n} m_i \cdot \mathbf{v}_i$$ Conservation laws facilitate solving complex problems involving several interacting objects by reducing the number of equations required.

Relative Velocities in Collisions

Understanding relative velocities is essential in collision problems. The relative velocity between two objects before and after collision helps in determining the nature of the collision (elastic or inelastic) and the post-collision velocities. $$\mathbf{v}_{\text{relative}} = \mathbf{v}_1 - \mathbf{v}_2$$

Impulse-Momentum Theorem

The impulse-momentum theorem links the impulse applied to an object to its change in momentum: $$J = \Delta p$$ This theorem is instrumental in analyzing scenarios where forces act over time intervals, such as collisions and impact events.

Practical Examples

Collision of Two Cars: Calculating post-collision velocities using mass and initial velocities to ensure momentum conservation.
Recoil of a Gun: Determining the gun's recoil velocity when a bullet is fired by conserving momentum between the gun and bullet.
Rocket Propulsion: Analyzing how expelled gases result in the rocket's motion by conserving momentum.

Mathematical Problems and Solutions

Example 1: Two ice skaters push off from one another. Skater A has a mass of 50 kg and moves at 2 m/s. If Skater B moves at 3 m/s, find the mass of Skater B.

Solution:

Using conservation of momentum: $$m_A \cdot v_A + m_B \cdot v_B = 0$$ Since they push off each other, their momenta are equal and opposite: $$m_A \cdot v_A = m_B \cdot (-v_B)$$ $$50 \cdot 2 = m_B \cdot 3$$ $$m_B = \frac{100}{3} \approx 33.33 \text{ kg}$$

Advanced Concepts

Momentum in Non-Inertial Frames

While momentum conservation is straightforward in inertial frames, analyzing it in non-inertial frames (accelerating or rotating) introduces complexities. In such frames, pseudo-forces must be considered to account for the acceleration, thereby modifying the momentum conservation equations. Understanding these adjustments is crucial for accurately describing motion in rotating systems or accelerating reference frames.

Relativistic Momentum

At speeds approaching the speed of light, classical momentum definitions fail. Relativistic momentum incorporates the effects of Special Relativity: $$p = \gamma m v$$ where: $$\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$ Here, **c** is the speed of light. This formulation ensures momentum conservation holds true at relativistic speeds, aligning with Einstein's theories.

Angular Momentum Conservation

Extending beyond linear momentum, angular momentum considers rotational motion. The conservation of angular momentum states that if no external torque acts on a system, the total angular momentum remains constant. This principle explains phenomena like a figure skater spinning faster when pulling in their arms and the stability of planetary orbits.

Momentum in Quantum Mechanics

In quantum mechanics, momentum conservation persists but is described through momentum operators and wavefunctions. The Heisenberg Uncertainty Principle introduces limits to simultaneously knowing position and momentum, but conservation laws still apply in particle interactions and reactions at the quantum level.

Deep Dive: Derivation Using Newtonian Mechanics

To rigorously derive momentum conservation, consider a system of two interacting particles with masses **m₁** and **m₂**, velocities **v₁** and **v₂** before collision, and **u₁** and **u₂** after collision. Applying Newton's Third Law: $$F_{12} = -F_{21}$$ Over the collision time **Δt**, the impulses are: $$J_{12} = F_{12} \cdot \Delta t = \Delta p_1$$ $$J_{21} = F_{21} \cdot \Delta t = \Delta p_2$$ Since **F₁₂** = -**F₂₁**, it follows that: $$\Delta p_1 + \Delta p_2 = 0$$ Thus, the total momentum remains unchanged: $$m₁v₁ + m₂v₂ = m₁u₁ + m₂u₂$$

Central Forces and Momentum Conservation

In systems where forces are central (acting along the line connecting centers of two particles), angular momentum conservation often accompanies linear momentum conservation. Analyzing such systems requires understanding both linear and angular momentum dynamics, especially in celestial mechanics and particle physics.

Perfectly Elastic and Inelastic Collisions: Energy Considerations

While momentum is conserved in all collisions, kinetic energy conservation varies:

Perfectly Elastic: Both momentum and kinetic energy are conserved. Objects rebound without energy loss.
Perfectly Inelastic: Momentum is conserved, but kinetic energy is not. Objects stick together, maximizing energy loss.
Inelastic: Momentum is conserved; some kinetic energy is lost to other forms like heat or sound.

Understanding these distinctions is crucial for accurately modeling collision outcomes.

Momentum Conservation in Fluids

In fluid dynamics, momentum conservation is expressed through the Navier-Stokes equations, which account for the transfer of momentum within fluid substances. These principles are vital for modeling weather patterns, ocean currents, and aerodynamics.

Conservation Laws and Noether’s Theorem

Noether’s Theorem connects conservation laws to symmetries in physics. Momentum conservation is linked to the homogeneity of space, meaning that the laws of physics are the same everywhere. This deep theoretical foundation underscores the universality of momentum conservation across diverse physical systems.

Momentum in Colliding Particle Physics

In high-energy physics, momentum conservation is essential in analyzing particle collisions in accelerators. It helps in determining the properties of subatomic particles and understanding fundamental interactions.

Momentum Distribution in Statistical Mechanics

Statistical mechanics uses momentum distribution functions to describe the behavior of particles in a system at thermodynamic equilibrium. These distributions are crucial for predicting macroscopic properties like temperature and pressure from microscopic dynamics.

Practical Problem Solving in Advanced Scenarios

Two-Dimensional Collisions: Solving momentum conservation problems where objects collide at angles, requiring the application of vector components.
Variable Mass Systems: Analyzing systems where mass changes over time, such as rockets expelling fuel, using principles of momentum conservation.
Impulse Calculations in Sports: Determining the force exerted by athletes by analyzing changes in momentum during actions like hitting or throwing.

Interdisciplinary Connections

Momentum conservation intersects with various fields:

Engineering: Designing mechanical systems and structures to withstand dynamic forces.
Biology: Understanding biomechanics and the motion of living organisms.
Economics: Drawing analogies between momentum in physics and momentum in financial markets.
Chemistry: Analyzing molecular collisions and reaction dynamics.

These connections highlight the versatility and broad applicability of momentum conservation principles.

Numerical Methods and Computational Simulations

Advanced analysis often employs numerical methods and computer simulations to solve complex momentum conservation problems. Techniques like finite element analysis (FEA) and computational fluid dynamics (CFD) allow for precise modeling of systems where analytical solutions are intractable.

Experimental Measurements of Momentum Conservation

Laboratory experiments validate momentum conservation through controlled collisions, projectile motion studies, and force measurements. Techniques include high-speed videography, motion sensors, and force transducers to accurately capture and analyze momentum changes.

Comparison Table

Aspect	Elastic Collisions	Inelastic Collisions
Kinetic Energy	Conserved	Not conserved
Momentum	Conserved	Conserved
Post-Collision Behavior	Objects bounce apart	Objects may stick together
Applications	Ideal gas particles, atomic collisions	Car crashes, ball in clay

Summary and Key Takeaways

State conservation of momentum ensures total momentum remains constant in closed systems.
Applicable to various collision types, with differing energy conservation.
Advanced concepts include relativistic momentum and angular momentum conservation.
Integral in multiple disciplines, from engineering to quantum mechanics.
Fundamental for solving complex dynamic problems and understanding motion.

Examiner Tip

Tips

To master momentum conservation, practice breaking down complex problems into smaller, manageable parts. Use the mnemonic MV Equals PV to remember that Momentum equals mass times velocity. Additionally, always sketch the scenario and identify all forces and objects involved before diving into calculations. This visual aid can help prevent common mistakes and ensure a comprehensive understanding of the problem.

Did You Know

Did you know that the principle of momentum conservation was first formulated by Sir Isaac Newton in his groundbreaking work, the Philosophiæ Naturalis Principia Mathematica? Additionally, momentum conservation is not only pivotal in classical mechanics but also plays a crucial role in modern physics, including particle physics and cosmology. For instance, the collision of particles in the Large Hadron Collider relies on momentum conservation to predict and analyze the outcomes of high-energy experiments.

Common Mistakes

Students often confuse mass and weight when calculating momentum, leading to incorrect results. Remember, momentum depends on mass (kg) and velocity (m/s), not weight. Another common mistake is ignoring the direction of velocity in vector calculations, which is essential for conserving momentum in different directions. Lastly, in collision problems, failing to account for all objects involved can result in incomplete momentum calculations. Always ensure that all interacting bodies are included in your analysis.

FAQ

What is the difference between elastic and inelastic collisions?

Elastic collisions conserve both momentum and kinetic energy, while inelastic collisions only conserve momentum. In inelastic collisions, some kinetic energy is transformed into other forms of energy, such as heat or deformation.

Can momentum be conserved in an open system?

Yes, but only when considering all external influences. In an open system where external forces act, the total momentum changes. However, within the system itself, if no external forces are present, momentum is conserved.

How does the conservation of momentum apply to rocket propulsion?

Rocket propulsion relies on momentum conservation by expelling gas backward at high speed, which propels the rocket forward. The momentum gained by the rocket is equal and opposite to the momentum of the expelled gas.

Why is momentum considered a vector quantity?

Momentum is a vector because it has both magnitude and direction, which are essential for accurately describing the motion and interactions of objects in different directions.

How is momentum conserved in collisions involving more than two objects?

In collisions with multiple objects, the total momentum is the vector sum of the individual momenta of all objects involved before and after the collision. As long as no external forces act on the system, this total momentum remains constant.

1. Capacitance

1.1 Energy Stored in a Capacitor

1.1.1 Determine electric potential energy stored in a capacitor from the area under the potential–charge g

1.1.2 Recall and use W = ½QV = ½CV²

1.2 Discharging a Capacitor

1.2.1 Analyze graphs of potential difference, charge, and current during capacitor discharge

1.2.2 Recall and use τ = RC for the time constant during discharge

1.2.3 Use equations of the form x = x₀e^(-t / RC) for current, charge, or potential during discharge

1.3 Capacitors and Capacitance

1.3.1 Define capacitance for isolated spherical conductors and parallel plate capacitors

1.3.2 Recall and use C = Q / V

1.3.3 Derive formulae for combined capacitance of capacitors in series and parallel

1.3.4 Use capacitance formulae for capacitors in series and parallel

2. Nuclear Physics

2.1 Mass Defect and Nuclear Binding Energy

2.1.1 Sketch the variation of binding energy per nucleon with nucleon number

2.1.2 Explain nuclear fusion and nuclear fission

2.1.3 Calculate the energy released in nuclear reactions using E = c²∆m

2.1.4 Understand the equivalence between energy and mass (E = mc²)

2.1.5 Define and use the terms mass defect and binding energy

2.2 Radioactive Decay

2.2.1 Understand that fluctuations in count rate provide evidence for random decay

2.2.2 Understand that radioactive decay is spontaneous and random

2.2.3 Define activity and decay constant, and recall A = λN

2.2.4 Define half-life

2.2.5 Use λ = 0.693 / t₁/₂ for the half-life relationship

2.2.6 Understand exponential decay and use the relationship x = x₀e^(-λt)

3. Kinematics

3.1 Equations of Motion

3.1.1 Define and use distance, displacement, speed, velocity, acceleration

3.1.2 Use graphical methods to represent motion

3.1.3 Determine displacement from velocity–time graph

3.1.4 Determine velocity using gradient of displacement–time graph

3.1.5 Determine acceleration using gradient of velocity–time graph

3.1.6 Derive equations for uniformly accelerated motion

3.1.7 Solve problems of uniformly accelerated motion

3.1.8 Describe experiment to determine acceleration due to gravity

3.1.9 Explain motion with uniform velocity and perpendicular acceleration

4. Deformation of Solids

4.1 Stress and Strain

4.1.1 Understand and use the terms load, extension, compression, and limit of proportionality

4.1.2 Recall and use Hooke’s law

4.1.3 Recall and use the formula for spring constant k = F / x

4.1.4 Define and use terms stress, strain, and Young's modulus

4.1.5 Describe an experiment to determine Young’s modulus

4.1.6 Understand deformation caused by tensile or compressive forces

4.2 Elastic and Plastic Behaviour

4.2.1 Understand elastic and plastic deformation and elastic limit

4.2.2 Understand area under the force–extension graph as work done

4.2.3 Calculate elastic potential energy using EP = ½kx²

5. Gravitational Fields

5.1 Gravitational Field of a Point Mass

5.1.1 Understand that g is approximately constant for small height changes near Earth's surface

5.1.2 Recall and use g = GM / r²

5.1.3 Derive and use g = GM / r² for gravitational field strength

5.2 Gravitational Potential

5.2.1 Define gravitational potential as work done per unit mass in bringing a test mass from infinity

5.2.2 Use ϕ = -GM / r for gravitational potential due to a point mass

5.2.3 Use EP = -GMm / r for gravitational potential energy between two point masses

5.3 Gravitational Field

5.3.1 Define gravitational field as force per unit mass

5.3.2 Represent a gravitational field using field lines

5.4 Gravitational Force Between Point Masses

5.4.1 For point outside uniform sphere, treat mass as a point mass at its center

5.4.2 Analyze circular orbits in gravitational fields

5.4.3 Recall and use Newton's law of gravitation F = Gm₁m₂ / r²

6. Electricity

6.1 Resistance and Resistivity

6.1.1 Understand that the resistance of a thermistor decreases as temperature increases

6.1.2 Define resistance and recall and use V = IR

6.1.3 Sketch I–V characteristics for metallic conductor, semiconductor diode, and filament lamp

6.1.4 Explain that the resistance of a filament lamp increases as current increases

6.1.5 State Ohm’s law and recall and use R = ρL / A

6.1.6 Understand that the resistance of an LDR decreases as light intensity increases

6.2 Electric Current

6.2.1 Understand that an electric current is a flow of charge carriers

6.2.2 Understand that the charge on charge carriers is quantised

6.2.3 Recall and use Q = It

6.2.4 Use I = Anvq for current-carrying conductors

6.3 Potential Difference and Power

6.3.1 Define potential difference as energy transferred per unit charge

6.3.2 Recall and use V = W / Q

6.3.3 Recall and use P = VI, P = I²R, P = V² / R

7. D.C. Circuits

7.1 Practical Circuits

7.1.1 Recall and use the circuit symbols in the syllabus

7.1.2 Draw and interpret circuit diagrams using circuit symbols

7.1.3 Define electromotive force (e.m.f.) of a source as energy transferred per unit charge

7.1.4 Distinguish between e.m.f. and potential difference (p.d.)

7.1.5 Understand the effects of internal resistance on terminal potential difference

7.2 Kirchhoff’s Laws

7.2.1 Recall Kirchhoff’s first law: conservation of charge

7.2.2 Recall Kirchhoff’s second law: conservation of energy

7.2.3 Derive the formula for combined resistance of resistors in series using Kirchhoff’s laws

7.2.4 Use the formula for combined resistance of resistors in series

7.2.5 Derive the formula for combined resistance of resistors in parallel using Kirchhoff’s laws

7.2.6 Use the formula for combined resistance of resistors in parallel

7.2.7 Use Kirchhoff’s laws to solve simple circuit problems

7.3 Potential Dividers

7.3.1 Understand the principle of a potential divider circuit

7.3.2 Recall and use the principle of the potentiometer for comparing potential differences

7.3.3 Understand the use of a galvanometer in null methods

7.3.4 Explain the use of thermistors and LDRs in potential dividers to provide a potential dependent on te

8. Thermal Equilibrium

8.1 Thermal Energy Transfer

8.1.1 Understand that thermal energy transfers from higher to lower temperature

8.1.2 Understand that regions of equal temperature are in thermal equilibrium

9. Temperature Scales

9.1 Temperature Measurement

9.1.1 Understand physical properties that vary with temperature (e.g., density, gas volume, resistance)

9.1.2 Convert temperatures between Kelvin and Celsius, and recall T(K) = θ(°C) + 273.15

10. Magnetic Fields

10.1 Concept of a Magnetic Field

10.1.1 Understand that a magnetic field is a field of force produced by moving charges or permanent magnets

10.1.2 Represent a magnetic field using field lines

10.2 Force on a Current-Carrying Conductor

10.2.1 Understand that a force acts on a current-carrying conductor in a magnetic field

10.2.2 Recall and use F = BIL sin θ for the force on a current-carrying conductor in a magnetic field

10.2.3 Define magnetic flux density as the force per unit current per unit length on a wire at right angles

10.3 Force on a Moving Charge

10.3.1 Determine the direction of the force on a charge moving in a magnetic field

10.3.2 Recall and use F = BQv sin θ for the force on a moving charge in a magnetic field

10.3.3 Understand the origin of the Hall voltage and use the expression VH = BI / (ntq) for the Hall voltag

10.3.4 Understand the use of a Hall probe to measure magnetic flux density

10.4 Magnetic Fields Due to Currents

10.4.1 Sketch magnetic field patterns due to currents in a straight wire, flat coil, and solenoid

10.4.2 Understand that magnetic field in a solenoid is increased by a ferrous core

10.4.3 Explain the origin of forces between current-carrying conductors and determine the direction of the

10.5 Electromagnetic Induction

10.5.1 Define magnetic flux as the product of magnetic flux density and area perpendicular to flux directio

10.5.2 Recall and use Φ = BA for magnetic flux

10.5.3 Understand and use the concept of magnetic flux linkage

10.5.4 Explain experiments that show changing magnetic flux induces an e.m.f.

10.5.5 Recall and use Faraday’s and Lenz’s laws of electromagnetic induction

11. Specific Heat Capacity and Latent Heat

11.1 Specific Heat Capacity

11.1.1 Define and use specific heat capacity

11.2 Latent Heat

11.2.1 Define and use specific latent heat; distinguish between fusion and vaporisation

12. Dynamics

12.1 Momentum and Newton’s Laws of Motion

12.1.1 Mass resists change in motion

12.1.2 Recall F = ma and solve related problems

12.1.3 Define and use linear momentum

12.1.4 Force as rate of change of momentum

12.1.5 State and apply Newton’s laws

12.1.6 Describe weight as effect of gravitational field

12.1.7 Weight of an object is mass × gravitational acceleration

12.2 Non-uniform Motion

12.2.1 Understand frictional and drag forces

12.2.2 Describe motion in uniform gravitational field with air resistance

12.2.3 Objects moving against resistive force may reach terminal velocity

12.3 Linear Momentum and its Conservation

12.3.1 State conservation of momentum

12.3.2 Apply conservation of momentum to solve problems

12.3.3 For elastic collision, kinetic energy and relative speed are conserved

12.3.4 Momentum is conserved, but some kinetic energy may change

13. Medical Physics

13.1 Production and Use of Ultrasound

13.1.1 Understand that a piezo-electric crystal changes shape when a p.d. is applied and generates an e.m.f

13.1.2 Understand how ultrasound waves are generated and detected by a piezoelectric transducer

13.1.3 Understand how ultrasound reflection at boundaries provides diagnostic information about internal st

13.1.4 Define specific acoustic impedance as Z = ρc

13.1.5 Use the equation IR / I₀ = (Z₁ – Z₂)² / (Z₁ + Z₂)² for the intensity reflection coefficient

13.2 Production and Use of X-rays

13.2.1 Explain that X-rays are produced by electron bombardment of a metal target and calculate the minimum

13.2.2 Understand the use of X-rays in imaging internal body structures and the term 'contrast' in X-ray im

13.2.3 Recall and use I = I₀e^(-μx) for the attenuation of X-rays in matter

13.2.4 Understand how computed tomography (CT) produces 3D images by combining multiple 2D X-ray images

13.3 PET Scanning

13.3.1 Understand that a tracer contains radioactive nuclei and is introduced into the body for imaging

13.3.2 Understand that β+ decay is used in positron emission tomography (PET)

13.3.3 Understand that annihilation occurs when a particle interacts with its antiparticle, conserving mass

13.3.4 Explain how gamma-ray photons from annihilation are detected to create an image of tracer concentrat

14. Waves

14.1 Progressive Waves

14.1.1 Describe wave motion in ropes, springs, and ripple tanks

14.1.2 Understand and use the terms displacement, amplitude, phase difference, period, frequency, wavelengt

14.1.3 Use time-base and y-gain on a cathode-ray oscilloscope (CRO) to determine frequency and amplitude

14.1.4 Derive and use v = f λ for the wave equation

14.1.5 Recall and use v = f λ

14.1.6 Understand energy transfer by a progressive wave

14.1.7 Recall and use intensity = power / area, intensity ∝ (amplitude)²

14.2 Transverse and Longitudinal Waves

14.2.1 Compare transverse and longitudinal waves

14.2.2 Analyze and interpret graphical representations of transverse and longitudinal waves

14.3 Doppler Effect for Sound Waves

14.3.1 Understand the change in frequency when a sound source moves relative to a stationary observer

14.3.2 Use the expression f₀ = fₛ(v / (v ± vₛ)) for the observed frequency

14.4 Electromagnetic Spectrum

14.4.1 State all electromagnetic waves are transverse waves traveling at the same speed in free space

14.4.2 Recall the range of wavelengths from radio waves to γ-rays

14.4.3 Recall that wavelengths 400–700 nm are visible to the human eye

14.5 Polarisation

14.5.1 Understand that polarisation is a phenomenon associated with transverse waves

14.5.2 Recall and use Malus's law (I = I₀ cos²θ) to calculate intensity of a plane-polarised wave after pas

15. The Mole

15.1 The Mole

15.1.1 Understand that amount of substance is an SI base quantity with the unit mol

15.1.2 Use molar quantities where one mole contains Avogadro's constant of particles

16. Equation of State

16.1 Ideal Gas Law

16.1.1 Understand that a gas obeying pV ∝ T is an ideal gas

16.1.2 Recall and use the equation pV = nRT for an ideal gas

16.1.3 Recall pV = NkT, where N = number of molecules and k is Boltzmann constant

16.1.4 Use the relationship k = R / NA

17. Kinetic Theory of Gases

17.1 Kinetic Theory

17.1.1 State the basic assumptions of the kinetic theory of gases

17.1.2 Explain pressure exerted by a gas due to molecular movement

17.1.3 Derive and use pV = (3/2)NmkT for pressure of a gas

17.1.4 Compare pV = (3/2)NmkT with pV = NkT to deduce average kinetic energy of a molecule

18. Particle Physics

18.1 Atoms, Nuclei and Radiation

18.1.1 Infer the existence and small size of the nucleus from α-particle scattering

18.1.2 Describe the nuclear atom model: protons, neutrons, and electrons

18.1.3 Distinguish between nucleon number and proton number

18.1.4 Understand isotopes as forms of the same element with different neutrons

18.1.5 Use the notation AZX for nuclides representation

18.1.6 Understand conservation of nucleon number and charge in nuclear processes

18.1.7 Describe the composition, mass, and charge of α-, β-, and γ-radiations

18.1.8 Understand antiparticles and positrons as antiparticles of electrons

18.1.9 State that (electron) antineutrinos are produced during β- decay and neutrinos during β+ decay

18.1.10 Represent α- and β-decay using radioactive decay equations

18.1.11 Use unified atomic mass unit (u) for mass

18.2 Fundamental Particles

18.2.1 Understand quarks as fundamental particles with six flavours

18.2.2 Understand hadrons as baryons (three quarks) or mesons (one quark and one antiquark)

18.2.3 Describe changes in quark composition during β- and β+ decay

18.2.4 Describe protons and neutrons as composed of quarks

18.2.5 Recall and use the charge of each quark and its respective antiquark

18.2.6 Recall electrons and neutrinos as fundamental particles (leptons)

19. Internal Energy

19.1 Internal Energy

19.1.1 Understand that internal energy is the sum of kinetic and potential energies of molecules in a syste

19.1.2 Relate rise in temperature to an increase in internal energy

20. Forces, Density and Pressure

20.1 Turning Effects of Forces

20.1.1 Weight acts at a single point: centre of gravity

20.1.2 Define and apply the moment of a force

20.1.3 Understand the concept of a couple

20.1.4 Define and apply torque of a couple

20.2 Equilibrium of Forces

20.2.1 State and apply the principle of moments

20.2.2 System in equilibrium has no resultant force or torque

20.2.3 Use a vector triangle to represent coplanar forces in equilibrium

20.3 Density and Pressure

20.3.1 Define and use density

20.3.2 Define and use pressure

20.3.3 Derive equation for hydrostatic pressure ∆p = ρg∆h

20.3.4 Use the equation ∆p = ρg∆h

20.3.5 Understand upthrust as the effect of hydrostatic pressure

20.3.6 Calculate upthrust using F = ρgV (Archimedes' principle)

21. Astronomy and Cosmology

21.1 Standard Candles

21.1.1 Understand luminosity as the total power radiated by a star

21.1.2 Recall and use the inverse square law F = L / (4πd²) for radiant flux intensity

21.1.3 Understand that an object with known luminosity is a standard candle

21.1.4 Use standard candles to determine distances to galaxies

21.2 Stellar Radii

21.2.1 Recall and use Wien’s displacement law λmax ∝ 1 / T to estimate the surface temperature of a star

21.2.2 Use Stefan–Boltzmann law L = 4πσr²T⁴

21.2.3 Use Wien’s law and Stefan–Boltzmann law to estimate the radius of a star

21.3 Hubble’s Law and Big Bang Theory

21.3.1 Understand redshift and the increase in wavelength of light from distant objects

21.3.2 Use ∆λ / λ = ∆f / f = v / c for redshift

21.3.3 Explain why redshift suggests the Universe is expanding

21.3.4 Recall and use Hubble’s law v = H₀d to explain the Big Bang theory

22. First Law of Thermodynamics

22.1 First Law

22.1.1 Recall and use W = p∆V for work done when the volume of a gas changes at constant pressure

22.1.2 Understand the difference between work done by a gas and work done on a gas

22.1.3 Recall and use the first law of thermodynamics: ∆U = q + W

23. Alternating Currents

23.1 Characteristics of Alternating Currents

23.1.1 Understand and use terms period, frequency, and peak value for alternating current or voltage

23.1.2 Use x = x₀ sin(ωt) for sinusoidal alternating current or voltage

23.1.3 Recall that the mean power in a resistive load is half the maximum power for sinusoidal current

23.1.4 Distinguish between root-mean-square (r.m.s.) and peak values and recall I₀r.m.s = I₀ / √2 and V₀r.m

23.2 Rectification and Smoothing

23.2.1 Distinguish graphically between half-wave and full-wave rectification

23.2.2 Explain the use of a single diode for half-wave rectification

23.2.3 Explain the use of four diodes (bridge rectifier) for full-wave rectification

23.2.4 Analyze the effect of a capacitor in smoothing, including the effect of capacitance and load resista

24. Superposition

24.1 Stationary Waves

24.1.1 Explain and use the principle of superposition

24.1.2 Understand experiments demonstrating stationary waves using microwaves, stretched strings, and air c

24.1.3 Explain the formation of stationary waves using graphical methods and identify nodes and antinodes

24.1.4 Determine wavelength from the positions of nodes or antinodes

24.2 Diffraction

24.2.1 Explain diffraction and show understanding of related experiments

24.2.2 Understand the effect of gap width relative to wavelength in diffraction experiments

24.3 Interference

24.3.1 Understand the terms interference and coherence

24.3.2 Demonstrate two-source interference using water waves, sound, light, and microwaves

24.3.3 Understand the conditions required for two-source interference fringes

24.3.4 Recall and use λ = ax / D for double-slit interference with light

24.4 Diffraction Grating

24.4.1 Recall and use d sin θ = nλ for diffraction grating calculations

24.4.2 Describe the use of a diffraction grating to determine the wavelength of light

25. Simple Harmonic Oscillations

25.1 Simple Harmonic Motion

25.1.1 Understand and use terms displacement, amplitude, period, frequency, angular frequency, and phase di

25.1.2 Understand that simple harmonic motion occurs when acceleration is proportional to displacement from

25.1.3 Use a = -ω²x and recall x = x₀ sin(ωt) as the solution

25.1.4 Use v = v₀ cos(ωt) and v = ±ω√(x₀² - x²) for velocity

26. Energy in Simple Harmonic Motion

26.1 Energy in SHM

26.1.1 Describe the interchange between kinetic and potential energy during SHM

26.1.2 Recall and use E = ½mω²x₀² for the total energy of a system undergoing SHM

27. Quantum Physics

27.1 Energy and Momentum of a Photon

27.1.1 Understand that electromagnetic radiation has a particulate nature

27.1.2 Recall and use E = hf for the energy of a photon

27.1.3 Use the electronvolt (eV) as a unit of energy

27.1.4 Understand that a photon has momentum and use p = E / c for photon momentum

27.2 Photoelectric Effect

27.2.1 Understand that photoelectrons are emitted from a metal surface when illuminated by electromagnetic

27.2.2 Understand and use the terms threshold frequency and threshold wavelength

27.2.3 Explain photoelectric emission in terms of photon energy and work function

27.2.4 Recall and use hf = Φ + ½mv² for the maximum kinetic energy of photoelectrons

27.2.5 Explain why the maximum kinetic energy of photoelectrons is independent of intensity, while current

27.3 Wave-Particle Duality

27.3.1 Understand that the photoelectric effect shows the particulate nature of light, while diffraction sh

27.3.2 Describe and interpret electron diffraction as evidence for the wave nature of particles

27.3.3 Understand the de Broglie wavelength as the wavelength associated with a moving particle

27.3.4 Recall and use λ = h / p for de Broglie wavelength

27.4 Energy Levels in Atoms and Line Spectra

27.4.1 Understand discrete electron energy levels in atoms (e.g., atomic hydrogen)

27.4.2 Understand the appearance and formation of emission and absorption line spectra

27.4.3 Recall and use hf = E₁ – E₂ for the energy difference between electron energy levels

28. Damped and Forced Oscillations, Resonance

28.1 Damped Oscillations

28.1.1 Understand that damping occurs due to resistive forces

28.1.2 Understand the types of damping: light, critical, and heavy damping

28.2 Resonance

28.2.1 Understand that resonance occurs when an oscillating system is forced to oscillate at its natural fr

29. Work, Energy and Power

29.1 Energy Conservation

29.1.1 Understand the concept of work, and recall W = F × s

29.1.2 Apply the principle of conservation of energy

29.1.3 Understand efficiency as useful energy output / total energy input

29.1.4 Use efficiency concept to solve problems

29.1.5 Define power as work done per unit time

29.1.6 Solve problems using P = W / t

29.1.7 Derive P = Fv and solve problems

29.2 Gravitational Potential Energy and Kinetic Energy

29.2.1 Derive ∆EP = mg∆h for gravitational potential energy changes

29.2.2 Recall and use ∆EP = mg∆h for gravitational potential energy changes

29.2.3 Derive EK = ½mv² for kinetic energy

29.2.4 Recall and use EK = ½mv² for kinetic energy

30. Electric Fields

30.1 Electric Fields and Field Lines

30.1.1 Understand that an electric field is a field of force and define it as force per unit positive charg

30.1.2 Represent an electric field using field lines

30.2 Uniform Electric Fields

30.2.1 Recall and use E = ∆V / ∆d to calculate electric field strength between charged parallel plates

30.2.2 Describe the effect of a uniform electric field on the motion of charged particles

30.3 Electric Force Between Point Charges

30.3.1 Understand that, for a point outside a spherical conductor, the charge may be considered a point mas

30.3.2 Recall and use Coulomb’s law F = Q₁Q₂ / (4πε₀r²) for the force between two point charges

30.4 Electric Field of a Point Charge

30.4.1 Recall and use E = Q / (4πε₀r²) for the electric field strength due to a point charge

30.5 Electric Potential

30.5.1 Define electric potential as work done per unit positive charge in bringing a test charge from infin

30.5.2 Recall and use V = Q / (4πε₀r) for electric potential due to a point charge

30.5.3 Understand how electric potential leads to electric potential energy of two point charges and use EP

31. Physical Quantities and Units

31.1 Physical Quantities

31.1.1 Physical quantities consist of magnitude and unit

31.1.2 Estimate physical quantities from the syllabus

31.2 SI Units

31.2.1 SI base units: mass, length, time, current, temperature

31.2.2 Express derived units from SI base units

31.2.3 Check homogeneity of physical equations

31.2.4 Use prefixes (pico, nano, micro, kilo, etc.)

31.3 Errors and Uncertainties

31.3.1 Understand systematic and random errors

31.3.2 Assess uncertainty in derived quantities

31.4 uniform Electric Fields

31.4.1 Distinguish between precision and accuracy

31.5 Scalars and Vectors

31.5.1 Difference between scalar and vector quantities

31.5.2 Add and subtract coplanar vectors

31.5.3 Represent a vector as components

32. Motion in a Circle

32.1 Centripetal Acceleration

32.1.1 Understand centripetal acceleration and its role in circular motion

32.1.2 Recall and use a = rω² and a = v² / r

32.1.3 Recall and use F = mrω² and F = mv² / r

32.1.4 Understand force causing centripetal acceleration is always perpendicular to motion

32.2 Kinematics of Uniform Circular Motion

32.2.1 Recall and use ω = 2π / T and v = rω

32.2.2 Understand and use angular speed

32.2.3 Define and express angular displacement in radians

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State Conservation of Momentum

Introduction