Recall and use the equation for kinetic energy E_k = ½ m v^2

1 Motion, Forces & Energy (Core)

1.1 Physical Quantities, Units & Symbols

Quantity	Symbol	SI Unit	Typical IGCSE Range
Distance / displacement	s, Δs	metre (m)	0.01 m – 200 m
Time	t, Δt	second (s)	0.1 s – 600 s
Speed / velocity	v, u, v_f	metre per second (m s⁻¹)	0.5 m s⁻¹ – 30 m s⁻¹
Acceleration	a	metre per second squared (m s⁻²)	0.1 m s⁻² – 10 m s⁻²
Mass	m	kilogram (kg)	0.01 kg – 100 kg
Force	F	newton (N)	0.5 N – 200 N
Momentum	p	kilogram metre per second (kg m s⁻¹)	0.1 kg m s⁻¹ – 500 kg m s⁻¹
Kinetic energy	E_k	joule (J)	0.1 J – 10⁴ J
Work / energy	W, E	joule (J)	1 J – 10⁵ J
Power	P	watt (W)	1 W – 5000 W
Pressure	p	pascal (Pa)	10³ Pa – 10⁶ Pa

1.2 Vectors & Resultants

• Vectors have magnitude and direction; scalars have magnitude only.
• Resultant of two perpendicular vectors A and B: \[ R=\sqrt{A^{2}+B^{2}} \]
• Graphical method – tip‑to‑tail (parallelogram) construction.

1.3 Distance‑time & Speed‑time Graphs

• Distance‑time (s‑t): slope = speed; area = not used.
• Speed‑time (v‑t): slope = acceleration; area = distance travelled.
• Horizontal line → constant speed; straight line through origin → constant acceleration.

1.4 Equations of Motion (constant acceleration)

\[ \begin{aligned} v &= u + a t \\ s &= u t + \tfrac12 a t^{2} \\ v^{2} &= u^{2} + 2 a s \end{aligned} \]

Where u = initial speed, v = final speed, s = displacement.

1.5 Newton’s Laws of Motion

1. Law of Inertia – an object remains at rest or in uniform motion unless acted on by a net force.
2. Second law – F = m a. Units: N = kg·m s⁻².
3. Third law – For every action there is an equal and opposite reaction.

1.6 Momentum & Impulse (Supplement)

Momentum:

\[ p = m v \]

Impulse (change in momentum):

\[ J = F\Delta t = \Delta p = m\Delta v \]

Relation to kinetic energy (useful when only momentum is known):

\[ E_k = \frac{p^{2}}{2m} \]

1.7 Kinetic Energy

Recall the kinetic‑energy equation:

\[ E_k = \frac12\,m\,v^{2} \]

Derivation (work‑energy theorem)

\[ W = \int_{0}^{s}F\,\mathrm{d}s = \int_{0}^{v} m a\,\frac{v}{a}\,\mathrm{d}v = \int_{0}^{v} m v\,\mathrm{d}v = \tfrac12 m v^{2} \]

Thus the net work done on an object equals its change in kinetic energy.

Worked Example

Problem: A 0.15 kg tennis ball leaves the racket at 35 m s⁻¹. Find its kinetic energy.

\[ E_k = \tfrac12 \times 0.15 \times (35)^{2} = 0.075 \times 1225 = 91.9\ \text{J} \approx 92\ \text{J} \]

Common Mistakes

• Forgetting to square the speed.
• Using grams instead of kilograms.
• Omitting the factor ½.
• Confusing kinetic energy (J) with momentum (kg m s⁻¹).

1.8 Work, Energy Transfer & Power

• Work (constant force): \(W = F s\) (J)
• Work‑energy theorem: \(W_{\text{net}} = \Delta E_k\)
• Power: \(P = \dfrac{W}{t} = \dfrac{\Delta E}{t}\) (W)

Worked Example – Pulling a Block

A 5 kg block is pulled across a frictionless floor by a 20 N force over 3 m, starting from rest. Find its final speed.

\[ W = Fs = 20 \times 3 = 60\ \text{J} \] \[ \Delta E_k = 60 = \tfrac12 \times 5 \times v^{2} \;\Rightarrow\; v^{2}=24\;\Rightarrow\; v = 4.9\ \text{m s}^{-1} \]

1.9 Pressure

Pressure is force per unit area:

\[ p = \frac{F}{A}\qquad\text{(Pa)} \]

Example

A rider exerts 800 N on a tyre with contact area 0.02 m².

\[ p = \frac{800}{0.02}=4.0\times10^{4}\ \text{Pa}=0.4\ \text{bar} \]

1.10 Energy Stores & Conversions

Store	Symbol	Formula (where applicable)	Typical Example
Kinetic	E_k	\(\tfrac12 m v^{2}\)	Moving car
Gravitational‑potential	E_g	\(m g h\)	Water behind a dam
Elastic	E_e	\(\tfrac12 k x^{2}\)	Stretched spring
Thermal (internal)	E_{th}	\(m c \Delta\theta\)	Hot water
Chemical	E_c	—	Battery, food
Electrostatic	E_{es}	\(\tfrac12 C V^{2}\)	Charged capacitor
Nuclear	E_n	—	Radioactive decay

Energy is conserved; it can be transferred between stores (e.g., a falling object converts E_g to E_k).

2 Thermal Physics (Core)

2.1 Particle Model of Matter

• Solids: particles tightly packed, vibrate about fixed positions.
• Liquids: particles close but can move past one another.
• Gases: particles far apart, move freely.

2.2 Temperature & Temperature Scales

• Thermometers measure temperature (θ).
• Conversion: \(\displaystyle T\ (\text{K}) = \theta\ (\!^\circ\text{C}) + 273\).

2.3 Specific Heat Capacity

Energy required to raise the temperature of 1 kg of a substance by 1 K:

\[ Q = m c \Delta\theta \]

where \(Q\) is heat energy (J), \(c\) is specific heat capacity (J kg⁻¹ K⁻¹).

Worked Example

How much energy is needed to heat 0.5 kg of aluminium (c = 900 J kg⁻¹ K⁻¹) from 20 °C to 80 °C?

\[ Q = 0.5 \times 900 \times (80-20) = 0.5 \times 900 \times 60 = 27\,000\ \text{J} \]

2.4 Latent Heat (Phase Change)

\[ Q = m L \]

• L_f – latent heat of fusion (solid↔liquid).
• L_v – latent heat of vaporisation (liquid↔gas).

Example

Calculate the energy required to melt 0.2 kg of ice (L_f = 3.3 × 10⁵ J kg⁻¹).

\[ Q = 0.2 \times 3.3\times10^{5}=6.6\times10^{4}\ \text{J} \]

2.5 Thermal Expansion

Linear expansion: \(\displaystyle \Delta L = \alpha L_0 \Delta\theta\) (α = coefficient of linear expansion).

Area expansion: \(\displaystyle \Delta A = 2\alpha A_0 \Delta\theta\).

2.6 Heat Transfer Methods

• Conduction: direct particle collisions; rate \( \dot Q = \frac{kA}{d}\Delta\theta\).
• Convection: bulk movement of fluid; described qualitatively.
• Radiation: electromagnetic waves; power \(P = \varepsilon\sigma A T^{4}\).

3 Waves (Core)

3.1 Wave Terminology

• Wave: disturbance that transfers energy without permanent displacement of matter.
• Amplitude (A), wavelength (λ), frequency (f), period (T), speed (v).

Fundamental relation:

\[ v = f\lambda = \frac{\lambda}{T} \]

3.2 Reflection & Refraction

• Reflection: angle of incidence = angle of reflection.
• Refraction: change of direction when wave passes into a medium with different speed; Snell’s law \(\displaystyle n_1\sin\theta_1 = n_2\sin\theta_2\).

3.3 Diffraction

Significant when the obstacle size is comparable to the wavelength. Demonstrated with water waves or sound through a doorway.

3.4 Light – Mirrors & Lenses

• Plane mirror: image distance = object distance, laterally inverted.
• Concave mirror: focal length f, mirror equation \(\displaystyle \frac{1}{u}+\frac{1}{v}= \frac{1}{f}\).
• Convex lens: real image (if object beyond f) or virtual image (if within f); same lens formula.

3.5 Sound

• Longitudinal wave; speed in air ≈ 340 m s⁻¹ (depends on temperature).
• Pitch ∝ frequency; loudness ∝ amplitude.

3.6 Electromagnetic Spectrum (Supplement)

From low to high frequency: radio → microwave → infrared → visible → ultraviolet → X‑ray → gamma‑ray.

4 Electricity & Magnetism (Core)

4.1 Charge, Current & Potential Difference

• Charge (Q) measured in coulombs (C). 1 C = 6.25 × 10¹⁸ e⁻.
• Current (I) = charge flow per unit time: \(I = \dfrac{\Delta Q}{\Delta t}\) (A).
• Potential difference (V) = energy per unit charge: \(V = \dfrac{W}{Q}\) (V).

4.2 Resistance & Ohm’s Law

\[ V = I R \]

Resistivity: \(R = \rho \dfrac{L}{A}\) (Ω), where ρ is material resistivity.

4.3 Series & Parallel Circuits

• Series: \(R_{\text{eq}} = R_1+R_2+\dots\); same current, voltages add.
• Parallel: \(\displaystyle \frac{1}{R_{\text{eq}}}= \frac{1}{R_1}+ \frac{1}{R_2}+ \dots\); same voltage, currents add.

4.4 Electrical Power & Energy

\[ P = VI = I^{2}R = \frac{V^{2}}{R} \qquad E = Pt \]

Worked Example – Electric Kettle

Power = 2 kW, time = 3 min.

\[ E = P t = 2000 \times 180 = 3.6\times10^{5}\ \text{J} \]

4.5 Magnetic Fields

• Field lines emerge from north pole, enter south pole.
• Force on a moving charge: \(\displaystyle \mathbf{F}=q\mathbf{v}\times\mathbf{B}\).
• Force on a current‑carrying conductor: \(\displaystyle F = B I L \sin\theta\).

4.6 Electromagnetic Induction (Supplement)

Changing magnetic flux through a coil induces an emf (Faraday’s law):

\[ \mathcal{E} = -N\frac{\Delta \Phi}{\Delta t} \]

where \(\Phi = B A\) is magnetic flux.

5 Nuclear Physics (Core)

5.1 Radioactivity

• Alpha (α) particles: +2 e, heavy, low penetration.
• Beta (β) particles: electrons, moderate penetration.
• Gamma (γ) rays: high‑energy photons, highly penetrating.

5.2 Half‑Life

\[ N = N_0\left(\frac{1}{2}\right)^{t/t_{1/2}} \]

Useful for dating and calculating activity.

5.3 Nuclear Reactions & Energy Release

• Fission: heavy nucleus → lighter fragments + energy.
• Fusion: light nuclei combine → heavier nucleus + energy.

Energy released per nucleon ≈ 8 MeV, far greater than chemical reactions.

6 Space Physics (Core)

6.1 Universal Gravitation

\[ F = G\frac{m_1 m_2}{r^{2}} \qquad g = G\frac{M_{\earth}}{R_{\earth}^{2}} \approx 9.8\ \text{m s}^{-2} \]

6.2 Satellite Motion & Orbital Speed

\[ \frac{mv^{2}}{r} = G\frac{M_{\earth}m}{r^{2}} \;\Rightarrow\; v = \sqrt{\frac{G M_{\earth}}{r}} \]

6.3 Escape Velocity

\[ v_{\text{esc}} = \sqrt{\frac{2 G M_{\earth}}{R_{\earth}}}\approx 11.2\ \text{km s}^{-1} \]

7 Practical Skills (Across All Topics)

7.1 Experimental Design

• Identify variables: independent, dependent, controlled.
• Plan a method that isolates the effect of the independent variable.
• Include safety considerations (e.g., PPE, electrical hazards).

7.2 Uncertainty & Error Analysis

• Absolute uncertainty: ± value (e.g., ±0.01 m).
• Relative (fractional) uncertainty: \(\displaystyle \frac{\Delta x}{x}\).
• Propagation for multiplication/division: add relative uncertainties.

7.3 Graphical Interpretation

• Gradient = \(\Delta y/\Delta x\) → physical quantity (e.g., speed, resistance).
• Area under a curve → accumulated quantity (e.g., work, impulse).
• Best‑fit straight line → use two points or linear regression.

7.4 Common Laboratory Techniques

• Using stop‑watches and meter‑sticks – estimate reaction time and reading errors.
• Voltage and current measurement – correct use of voltmeter (parallel) and ammeter (series).
• Measuring mass – use of balance, tare function, and avoiding static charge.

Practice Questions (All Core Codes)

1. (1.7.1 Core) A 2 kg cart moves at 4 m s⁻¹. What is its kinetic energy?
2. (1.7.1 Core) A cyclist of mass 70 kg (including bike) travels at 6 m s⁻¹. Calculate the kinetic energy.
3. (1.7.1 Core) If a ball’s kinetic energy is 20 J and its mass is 0.2 kg, what is its speed?
4. (1.7.2 Supplement) Two objects have the same kinetic energy. One has a mass of 0.5 kg; the other has a mass of 2 kg. Which one is moving faster, and by what factor?
5. (1.2 Core) A 3 kg crate is pulled by a constant 15 N force over 5 m from rest. Find its final speed.
6. (1.3 Core) A 1500 W hair‑dryer runs for 5 minutes. How much energy does it use?
7. (1.4 Core) A force of 250 N is applied to a piston of area 0.005 m². Calculate the pressure exerted.
8. (2.3 Core) How much energy is required to heat 0.8 kg of water from 20 °C to 80 °C? (c_water = 4180 J kg⁻¹ K⁻¹)
9. (2.4 Core) 0.1 kg of ice at 0 °C melts completely. Find the heat absorbed. (L_f = 3.3 × 10⁵ J kg⁻¹)
10. (3.1 Core) A wave has a frequency of 50 Hz and a wavelength of 0.6 m. Calculate its speed.
11. (3.4 Core) An object is placed 0.15 m in front of a concave mirror with focal length 0.10 m. Determine the image distance and nature of the image.
12. (4.2 Core) A 10 Ω resistor is connected to a 12 V supply. Find the current and power dissipated.
13. (4.5 Core) A straight conductor 0.30 m long carries 5 A perpendicular to a magnetic field of 0.2 T. Find the magnetic force on the conductor.
14. (5.2 Core) A sample contains 8 × 10⁶ radioactive nuclei. Its half‑life is 2 h. How many nuclei remain after 6 h?
15. (6.2 Core) Calculate the orbital speed of a satellite 400 km above Earth’s surface. (R_earth = 6.37 × 10⁶ m, M_earth = 5.97 × 10²⁴ kg)

Answers

1. \(E_k = \tfrac12 \times 2 \times 4^{2}=16\ \text{J}\)
2. \(E_k = \tfrac12 \times 70 \times 6^{2}=1260\ \text{J}\)
3. \(v = \sqrt{\dfrac{2E_k}{m}} = \sqrt{\dfrac{2\times20}{0.2}} = \sqrt{200}\approx 14.1\ \text{m s}^{-1}\)
4. For equal \(E_k\): \(\dfrac{v_{0.5}}{v_{2}} = \sqrt{\dfrac{2}{0.5}} = 2\). The 0.5 kg object moves twice as fast.
5. \(W = Fs = 15 \times 5 = 75\ \text{J}\) \(\tfrac12 \times 3 \times v^{2}=75 \Rightarrow v^{2}=50 \Rightarrow v\approx7.1\ \text{m s}^{-1}\)
6. \(P=1500\ \text{W},\; t=5\ \text{min}=300\ \text{s}\) \(E = Pt = 1500 \times 300 = 4.5\times10^{5}\ \text{J}\)
7. \(p = \dfrac{250}{0.005}=5.0\times10^{4}\ \text{Pa}=0.5\ \text{bar}\)
8. \(Q = mc\Delta\theta = 0.8 \times 4180 \times (80-20)=2.0\times10^{6}\ \text{J}\)
9. \(Q = m L_f = 0.1 \times 3.3\times10^{5}=3.3\times10^{4}\ \text{J}\)
10. \(v = f\lambda = 50 \times 0.6 = 30\ \text{m s}^{-1}\)
11. Mirror formula: \(\frac{1}{u}+\frac{1}{v}= \frac{1}{f}\) with \(u = -0.15\ \text{m}, f = -0.10\ \text{m}\). \(\frac{1}{v}= \frac{1}{-0.10} - \frac{1}{-0.15}= -10 + 6.\overline{6}= -3.\overline{3}\) → \(v = -0.30\ \text{m}\). Image is real, inverted, and twice the size of the object.
12. \(I = V/R = 12/10 = 1.2\ \text{A}\) \(P = VI = 12 \times 1.2 = 14.4\ \text{W}\)
13. \(F = B I L = 0.2 \times 5 \times 0.30 = 0.30\ \text{N}\)
14. \(N = 8\times10^{6}\left(\frac12\right)^{6/2}=8\times10^{6}\left(\frac12\right)^{3}=8\times10^{6}\times\frac{1}{8}=1\times10^{6}\)
15. \(r = R_{\earth}+400\,000 = 6.77\times10^{6}\ \text{m}\) \(v = \sqrt{\frac{GM_{\earth}}{r}} = \sqrt{\frac{6.67\times10^{-11}\times5.97\times10^{24}}{6.77\times10^{6}}}\approx 7.7\times10^{3}\ \text{m s}^{-1}\)

Suggested Diagrams

• Vector addition (tip‑to‑tail) for two perpendicular forces.
• Distance‑time and speed‑time graphs illustrating constant speed and constant acceleration.
• Block being pushed: force F, displacement s, work W = Fs, resulting kinetic‑energy increase.
• Energy‑store Sankey diagram showing conversion from gravitational‑potential to kinetic to thermal.
• Ray diagram for a concave mirror (object beyond f, real inverted image).
• Circuit schematic showing series and parallel resistor combinations with a voltmeter and ammeter.
• Diagram of a satellite in circular orbit illustrating radius r and orbital speed v.