Understanding motion in a uniform gravitational field with air resistance is fundamental in physics, particularly within the study of dynamics. This topic is essential for students pursuing AS & A Level Physics (9702) as it bridges theoretical concepts with real-world applications. By exploring the interplay between gravitational forces and air resistance, students gain insight into projectile motion, terminal velocity, and the factors influencing an object's motion through a fluid medium.

Key Concepts

Uniform Gravitational Field

A uniform gravitational field is an idealization where the gravitational force experienced by an object is constant in both magnitude and direction. Near the Earth's surface, gravity can be approximated as uniform, with an acceleration due to gravity ($g$) of approximately $9.81 \, \text{m/s}^2$ downward.

The assumption of a uniform gravitational field simplifies the analysis of motion, allowing for the derivation of fundamental equations of kinematics and dynamics without accounting for variations in gravitational strength with altitude or location.

Air Resistance

Air resistance, also known as drag, is a force exerted by air against the motion of an object moving through it. Unlike gravity, which is constant for objects of the same mass, air resistance varies with factors such as the object's velocity, cross-sectional area, shape, and the density of the air.

The drag force ($F_d$) can be modeled using the equation:

$$ F_d = \frac{1}{2} C_d \rho A v^2 $$

where:

$C_d$ is the drag coefficient, dependent on the object's shape
$\rho$ is the air density
$A$ is the cross-sectional area perpendicular to the flow
$v$ is the velocity of the object relative to the air

Air resistance acts in the direction opposite to the object's velocity, affecting both the magnitude and direction of motion.

Equations of Motion with Air Resistance

When analyzing motion in a uniform gravitational field with air resistance, Newton's second law of motion is applied:

$$ \Sigma F = m a $$

For an object moving vertically, the forces acting on it are gravity ($F_g = m g$) downward and air resistance ($F_d$) upward if the object is descending, or downward if ascending.

The net force ($F_{net}$) equation becomes:

$$ F_{net} = m g - F_d $$ $$ m a = m g - \frac{1}{2} C_d \rho A v^2 $$

Solving for acceleration ($a$) gives:

$$ a = g - \frac{1}{2 m} C_d \rho A v^2 $$

This equation illustrates how air resistance reduces the acceleration of a falling object.

Terminal Velocity

Terminal velocity is achieved when the downward gravitational force equals the upward air resistance, resulting in zero net acceleration. At this point, the object continues to fall at a constant velocity.

Setting $F_{net} = 0$:

$$ m g = \frac{1}{2} C_d \rho A v_t^2 $$

Solving for terminal velocity ($v_t$):

$$ v_t = \sqrt{\frac{2 m g}{C_d \rho A}} $$

This equation shows that terminal velocity increases with mass and decreases with larger drag coefficients, increased air density, or larger cross-sectional areas.

Projectile Motion with Air Resistance

Projectile motion typically considers only uniform gravitational fields without air resistance. Introducing air resistance complicates the motion, as forces now depend on velocity, making the equations of motion nonlinear and often requiring numerical methods for solutions.

In two dimensions, the horizontal ($x$) and vertical ($y$) components of motion are affected differently:

Horizontal Motion: Air resistance causes deceleration proportional to velocity squared, leading to a gradual decrease in horizontal speed.
Vertical Motion: Air resistance opposes the motion, affecting both ascent and descent phases, and ultimately determining terminal velocity.

Kinematic Equations Modified for Air Resistance

Unlike in the absence of air resistance, kinematic equations cannot be directly applied. Instead, differential equations must be solved to account for the velocity-dependent drag force.

For vertical motion, the velocity as a function of time can be expressed as:

$$ v(t) = \sqrt{\frac{2 m g}{C_d \rho A}} \tanh\left(\sqrt{\frac{g C_d \rho A}{2 m}} t\right) $$

This equation shows how velocity approaches terminal velocity asymptotically as time increases.

Energy Considerations

Air resistance introduces non-conservative forces into the system, dissipating mechanical energy as heat. The work done against air resistance is given by:

$$ W = \int F_d \, ds = \int \frac{1}{2} C_d \rho A v^2 \, ds $$>

Since air resistance depends on velocity, the work done cannot be easily integrated without knowing the velocity as a function of displacement or time.

This energy loss affects the maximum height and range of projectiles compared to motion without air resistance.

Advanced Concepts

Mathematical Derivation of Motion Equations with Air Resistance

Deriving the equations of motion for an object in a uniform gravitational field with air resistance involves solving differential equations that account for velocity-dependent forces.

Starting with Newton's second law for vertical motion:

$$ m \frac{dv}{dt} = m g - \frac{1}{2} C_d \rho A v^2 $$>

Dividing both sides by $m$:

$$ \frac{dv}{dt} = g - \frac{C_d \rho A}{2 m} v^2 $$>

This is a separable differential equation. Rearranging terms:

$$ \frac{dv}{g - k v^2} = dt $$>

where $k = \frac{C_d \rho A}{2 m}$. Integrating both sides:

$$ \int \frac{dv}{g - k v^2} = \int dt $$>

The integral on the left side can be solved using partial fractions or standard integral formulas:

$$ \frac{1}{\sqrt{g k}} \tanh^{-1}\left(\sqrt{k/g} \, v \right) = t + C $$>

Solving for $v$ gives the velocity as a function of time:

$$ v(t) = \sqrt{\frac{g}{k}} \tanh\left(\sqrt{g k} \, t + C' \right) $$>

Applying initial conditions (e.g., $v(0) = 0$ for an object released from rest) allows determination of the constant $C'$. The final expression demonstrates how velocity asymptotically approaches terminal velocity as $t$ increases.

Numerical Methods for Solving Equations of Motion

Due to the nonlinear nature of the equations of motion with air resistance, analytical solutions are often limited to specific cases. For more complex scenarios, numerical methods such as Euler's method or the Runge-Kutta methods are employed to approximate solutions.

Euler's Method: A straightforward numerical approach where the next velocity and position are estimated using the current values and the derivatives.

Runge-Kutta Methods: More sophisticated techniques that provide higher accuracy by considering multiple intermediate steps within each time increment.

These methods are implemented using computational tools and software, allowing the simulation of projectile trajectories under various air resistance conditions.

Impact of Air Density on Motion

Air density ($\rho$) plays a significant role in determining the magnitude of air resistance. Higher air density increases the drag force, leading to greater energy dissipation and reduced terminal velocity.

Factors affecting air density include altitude, temperature, and atmospheric pressure. For example, at higher altitudes, lower air density results in decreased air resistance, allowing objects to achieve higher terminal velocities.

The equation for terminal velocity highlights this dependency:

$$ v_t = \sqrt{\frac{2 m g}{C_d \rho A}} $$>

Reducing air density ($\rho$) while keeping other factors constant results in an increase in terminal velocity ($v_t$).

Shapes and the Drag Coefficient

The drag coefficient ($C_d$) varies with the shape of the object and flow conditions. Streamlined bodies with shapes that facilitate smooth flow experience lower drag coefficients, while blunt or irregular shapes have higher $C_d$ values.

Examples of drag coefficients for different shapes include:

Sphere: $C_d \approx 0.47$
Cube: $C_d \approx 1.05$
Streamlined Body: $C_d \approx 0.04$

Engineering applications often prioritize shapes with low drag coefficients to minimize air resistance, enhancing performance and efficiency.

Interdisciplinary Connections

The study of motion in a uniform gravitational field with air resistance intersects various fields beyond physics:

Aerodynamics: Understanding drag forces is crucial in designing aircraft, automobiles, and sports equipment to optimize performance.
Environmental Science: Analyzing pollutant dispersion in the atmosphere involves accounting for air resistance and gravitational settling.
Biomechanics: Studying human motion, such as running or cycling, incorporates principles of air resistance to improve athletic performance.
Astrophysics: While space generally lacks air resistance, understanding drag forces is essential for studying atmospheric entry and re-entry of spacecraft.

These interdisciplinary applications demonstrate the broad relevance of motion analysis in various scientific and engineering domains.

Complex Problem-Solving: Example Problems

Problem 1: A skydiver of mass $80 \, \text{kg}$ jumps from an airplane. Assuming a drag coefficient $C_d = 1.0$, air density $\rho = 1.225 \, \text{kg/m}^3$, and cross-sectional area $A = 0.7 \, \text{m}^2$, calculate the terminal velocity.

Solution:

$$ v_t = \sqrt{\frac{2 m g}{C_d \rho A}} = \sqrt{\frac{2 \times 80 \times 9.81}{1.0 \times 1.225 \times 0.7}} \approx \sqrt{\frac{1569.6}{0.8575}} \approx \sqrt{1825.2} \approx 42.7 \, \text{m/s} $$>

The skydiver's terminal velocity is approximately $42.7 \, \text{m/s}$.

Problem 2: A ball is thrown vertically upward with an initial velocity of $20 \, \text{m/s}$. Considering air resistance with $C_d = 0.5$, $\rho = 1.225 \, \text{kg/m}^3$, $A = 0.05 \, \text{m}^2$, and mass $m = 0.2 \, \text{kg}$, determine the time to reach maximum height.

Solution:

The acceleration equation is:

$$ \frac{dv}{dt} = -g - \frac{1}{2 m} C_d \rho A v^2 $$>

At maximum height, $v = 0$. To find the time to reach this point, the differential equation must be integrated numerically due to the velocity-dependent term. Using numerical methods (e.g., Euler's method), the time can be approximated.

For simplicity, assuming negligible air resistance initially, the time to reach maximum height without air resistance is:

$$ t = \frac{v_0}{g} = \frac{20}{9.81} \approx 2.04 \, \text{seconds} $$>

Considering air resistance reduces this time slightly, a precise numerical solution would be required for an exact answer.

Comparison Table

Aspect	Motion without Air Resistance	Motion with Air Resistance
Forces Acting	Only gravity ($F = m g$)	Gravity and drag ($F = m g - \frac{1}{2} C_d \rho A v^2$)
Equations of Motion	Linear kinematic equations	Nonlinear differential equations
Terminal Velocity	Not applicable	Defined by $v_t = \sqrt{\frac{2 m g}{C_d \rho A}}$
Energy Considerations	Conservative forces only	Non-conservative drag force dissipates energy
Projectile Path	Asymmetrical due to velocity-dependent drag

Summary and Key Takeaways

Motion in a uniform gravitational field with air resistance involves both gravity and velocity-dependent drag forces.
Air resistance significantly affects acceleration, terminal velocity, and projectile trajectories.
Mathematical modeling requires solving nonlinear differential equations, often using numerical methods.
Understanding these concepts is crucial for applications in aerodynamics, biomechanics, and engineering.
Terminal velocity is a key concept, illustrating the balance between gravitational force and air resistance.

Examiner Tip

Tips

Remember the mnemonic "DAVE" to recall the factors affecting drag: Drag Coefficient, Air density, Velocity, and cross-sectional Area. To master differential equations involving air resistance, practice separating variables and integrating step-by-step. Utilize graphing calculators or software to visualize how velocity approaches terminal velocity over time. For exams, clearly state assumptions like uniform gravitational fields and specify when air resistance is being considered to avoid confusion.

Did You Know

Did you know that the shape of a bullet is meticulously designed to reduce air resistance, allowing it to achieve higher speeds and greater accuracy? Additionally, astronauts experience microgravity, where air resistance is virtually nonexistent, vastly different from the uniform gravitational fields we study on Earth. Another fascinating fact is that the concept of terminal velocity explains why skydivers deploy parachutes to increase drag and safely reduce their falling speed.

Common Mistakes

Students often assume that air resistance is constant, neglecting its dependence on velocity. For example, incorrectly applying $F_d = C_d$ instead of $F_d = \frac{1}{2} C_d \rho A v^2$ leads to errors in calculations. Another common mistake is ignoring the direction of air resistance, which always opposes motion, affecting both magnitude and direction. Additionally, students sometimes overlook the impact of air density changes with altitude, which is crucial for accurate terminal velocity calculations.

FAQ

What is terminal velocity?

Terminal velocity is the constant speed that an object reaches when the downward force of gravity is balanced by the upward force of air resistance, resulting in zero net acceleration.

How does air resistance affect projectile motion?

Air resistance introduces a force opposite to the direction of motion, making projectile paths asymmetrical and reducing both the maximum height and range compared to motion without air resistance.

Why are numerical methods needed to solve motion equations with air resistance?

Because air resistance makes the equations of motion nonlinear due to their dependence on velocity squared, analytical solutions become difficult or impossible, necessitating numerical approximation methods.

What factors influence the drag coefficient?

The drag coefficient is influenced by the shape of the object, flow conditions such as Reynolds number, and surface roughness. Streamlined shapes typically have lower drag coefficients.

How does air density vary with altitude?

Air density decreases with increasing altitude due to lower atmospheric pressure and temperature, which in turn reduces air resistance experienced by falling objects.

Can objects in space experience air resistance?

In the vacuum of space, air resistance is negligible because there is no atmosphere. However, during atmospheric entry or re-entry, objects encounter significant air resistance.

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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Describe Motion in Uniform Gravitational Field with Air Resistance

Introduction