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Mechanics

17 formulas · KCSE & CBC

Newton's Second Law

KCSE 2019, 2021, 2022

\(F = ma\)

F Force (N)

m mass (kg)

a acceleration (m/s²)

Force equals mass times acceleration. The foundation of classical mechanics.

Pro tip: 1 N = 1 kg·m/s². If mass is constant, doubling force doubles acceleration.

Mass (kg)

Acceleration (m/s²)

Force F (N)

—

A 5 kg box accelerates at 3 m/s². Find the force. F = ma = 5 × 3 = 15 N

Weight

KCSE 2018, 2020, 2022

\(W = mg\)

W Weight (N)

m mass (kg)

g 9.8 m/s²

The gravitational force acting on an object. Weight changes with gravity; mass does not.

Pro tip: On Moon g = 1.6 m/s². A 70 kg person weighs 686 N on Earth, 112 N on Moon.

Mass (kg)

g (m/s²)

Weight W (N)

—

Find weight of a 12 kg object on Earth. W = mg = 12 × 9.8 = 117.6 N

Kinetic Energy

KCSE 2017, 2019, 2021

\(KE = \frac{1}{2}mv^2\)

KE Kinetic Energy (J)

m mass (kg)

v velocity (m/s)

Energy possessed by a moving object. Doubling speed quadruples KE — this is why speed limits save lives.

Pro tip: 1 J = 1 kg·m²/s². A 1000 kg car at 30 m/s has 450,000 J of KE.

Mass (kg)

Velocity (m/s)

KE (Joules)

—

A 2 kg ball moves at 10 m/s. Find KE. KE = ½mv² = ½ × 2 × 10² = 100 J

Potential Energy

KCSE 2018, 2020

\(PE = mgh\)

PE Potential Energy (J)

m mass (kg)

g 9.8 m/s²

h height (m)

Energy stored due to an object's height. Converted to KE when the object falls.

Pro tip: A 1 kg book on a 1 m shelf stores 9.8 J. That energy is released when it falls.

Mass (kg)

g (m/s²)

Height (m)

PE (Joules)

—

A 3 kg stone is raised 5 m. Find PE. PE = mgh = 3 × 9.8 × 5 = 147 J

Work Done

KCSE 2017, 2019, 2022

\(W = Fd\)

W Work (J)

F Force (N)

d distance (m)

Energy transferred when a force moves an object. No movement = no work done, regardless of force applied.

Pro tip: Pushing a wall all day does zero work! Work requires displacement.

Force (N)

Distance (m)

Work W (Joules)

—

A force of 20 N moves an object 4 m. W = Fd = 20 × 4 = 80 J

Power

KCSE 2018, 2021

\(P = \frac{W}{t}\)

P Power (W)

W Work (J)

t time (s)

The rate of doing work or transferring energy. A more powerful engine does the same work faster.

Pro tip: 1 hp = 746 W. Your phone charger is ~20 W. A kettle is ~2000 W.

Work (J)

Time (s)

Power P (Watts)

—

100 J of work is done in 5 s. P = W/t = 100 ÷ 5 = 20 W

Momentum

KCSE 2017, 2020, 2022

\(p = mv\)

p momentum (kg·m/s)

m mass (kg)

v velocity (m/s)

Quantity of motion. Conserved in all collisions — the total before equals total after.

Pro tip: A slow truck can have more momentum than a fast car if it's much heavier.

Mass (kg)

Velocity (m/s)

Momentum p (kg·m/s)

—

A 1200 kg car travels at 15 m/s. p = mv = 1200 × 15 = 18000 kg·m/s

Density

KCSE 2019, 2021

\(\rho = \frac{m}{V}\)

ρ density (kg/m³)

m mass (kg)

V volume (m³)

How tightly matter is packed. Objects less dense than water float; denser objects sink.

Pro tip: Water = 1000 kg/m³. Ice = 917 kg/m³ (floats!). Iron = 7874 kg/m³.

Mass (kg)

Volume (m³)

Density ρ (kg/m³)

—

A block of mass 500 g has volume 250 cm³. ρ = m/V = 0.5 ÷ 0.00025 = 2000 kg/m³

Pressure

KCSE 2018, 2020, 2022

\(P = \frac{F}{A}\)

P Pressure (Pa)

F Force (N)

A Area (m²)

Force spread over an area. Smaller area = higher pressure with the same force.

Pro tip: Sharp knife edge = tiny area = massive pressure. Snow shoes = large area = low pressure.

Force (N)

Area (m²)

Pressure P (Pascals)

—

A 60 N force acts on 0.03 m² area. P = F/A = 60 ÷ 0.03 = 2000 Pa

Velocity (const. a)

KCSE 2017, 2019, 2021

\(v = u + at\)

v final velocity (m/s)

u initial velocity (m/s)

a acceleration (m/s²)

t time (s)

One of the SUVAT equations. Gives final velocity after constant acceleration over time.

Pro tip: If starting from rest, u = 0, so v = at. Free fall: a = g = 9.8 m/s².

Initial velocity u (m/s)

Acceleration (m/s²)

Time (s)

Final velocity v (m/s)

—

A car accelerates from rest at 2 m/s² for 8 s. v = u + at = 0 + 2 × 8 = 16 m/s

Displacement

KCSE 2018, 2021, 2022

\(s = ut + \frac{1}{2}at^2\)

s displacement (m)

u initial vel. (m/s)

t time (s)

a acceleration (m/s²)

Distance traveled under constant acceleration. Key SUVAT equation for projectile and free-fall problems.

Pro tip: For free fall from rest: s = ½gt². After 3 s, s = ½ × 9.8 × 9 = 44.1 m.

Initial velocity u (m/s)

Time (s)

Acceleration (m/s²)

Displacement s (m)

—

A ball is dropped from rest. Find distance after 4 s. s = ut + ½at² = 0 + ½ × 9.8 × 16 = 78.4 m

Acceleration (General)

KCSE 2018,2020,2023

\(a = \frac{v - u}{t}\)

a acceleration (m/s²)

v final velocity (m/s)

u initial velocity (m/s)

t time (s)

Rate of change of velocity. Positive acceleration = speeding up, negative = slowing down.

Pro tip: Deceleration is just negative acceleration. A car stopping from 20 m/s in 5 s has a = -4 m/s².

Final velocity v (m/s)

Initial velocity u (m/s)

Time t (s)

Acceleration a (m/s²)

—

Car speeds from 10 m/s to 30 m/s in 4 s. a = (30-10)/4 = 5 m/s²

Hookes Law

KCSE 2018,2020,2021

\(F = kx\)

F Force (N)

k spring constant (N/m)

x extension/compression (m)

Force needed to extend or compress a spring is proportional to displacement from equilibrium.

Pro tip: Spring constant k measures stiffness: higher k = stiffer spring.

Spring constant k (N/m)

Extension x (m)

Force F (N)

—

A spring with k = 200 N/m is stretched 0.1 m. F = 200 × 0.1 = 20 N

Spring Potential Energy

KCSE 2019,2021,2022

\(PE = \frac{1}{2}kx^2\)

PE potential energy (J)

k spring constant (N/m)

x displacement (m)

Energy stored in a stretched or compressed spring. Doubling stretch quadruples stored energy.

Pro tip: This energy is released when the spring returns to equilibrium.

Spring constant (N/m)

Displacement (m)

Spring PE (J)

—

k = 100 N/m, stretched 0.2 m. PE = ½ × 100 × 0.2² = 2 J

Friction Force

KCSE 2017,2019,2021

\(f = \mu N\)

f friction force (N)

μ coefficient of friction

N normal force (N)

Resistance force when two surfaces slide or attempt to slide. Static μ > kinetic μ.

Pro tip: On horizontal surface, N = mg. Friction does not depend on contact area!

Coefficient μ

Normal force N (N)

Friction Force f (N)

—

Box mass 10 kg on horizontal surface, μ = 0.4. N = mg = 98 N, f = 0.4 × 98 = 39.2 N

Centripetal Force

KCSE 2017,2019,2022

\(F_c = \frac{mv^2}{r}\)

F_c centripetal force (N)

m mass (kg)

v tangential speed (m/s)

r radius (m)

The inward force required to keep an object moving in a circle.

Pro tip: Centripetal means "center-seeking". No such thing as centrifugal force in inertial frames.

Mass (kg)

Speed v (m/s)

Radius r (m)

Centripetal Force (N)

—

500 kg car, radius 50 m, speed 20 m/s. F_c = 500×400/50 = 4000 N

Centripetal Acceleration

KCSE 2018,2020

\(a_c = \frac{v^2}{r}\)

a_c centripetal acceleration (m/s²)

v speed (m/s)

r radius (m)

Acceleration toward the center of a circular path.

Pro tip: a_c = ω²r (using angular velocity). At high speeds or small radii, a_c becomes huge.

Speed v (m/s)

Radius r (m)

Centripetal Acceleration (m/s²)

—

v = 10 m/s, r = 5 m. a_c = 100/5 = 20 m/s²

↗

Projectile Motion

3 formulas · KCSE & CBC

Max Height

KCSE 2018,2020,2022

\(H = \frac{u^2 \sin^2\theta}{2g}\)

H max height (m)

u initial velocity (m/s)

θ launch angle (°)

g 9.8 m/s²

Maximum vertical displacement reached by a projectile.

Pro tip: Max height is greatest at θ = 90° (straight up).

Initial speed u (m/s)

Launch angle θ (degrees)

g (m/s²)

Max Height H (m)

—

Launch speed 20 m/s, angle 30°. H = (400 × 0.25)/(2×9.8) = 5.1 m

Time of Flight

KCSE 2019,2021,2022

\(T = \frac{2u \sin\theta}{g}\)

T total time (s)

u initial velocity (m/s)

θ launch angle (°)

g 9.8 m/s²

Total time projectile spends in the air.

Pro tip: Time to max height = T/2 = u sinθ/g.

Initial speed u (m/s)

Launch angle θ (degrees)

g (m/s²)

Time of Flight T (s)

—

u = 30 m/s, θ = 40°. T = 2×30×0.643/9.8 = 3.94 s

Horizontal Range

KCSE 2017,2019,2021

\(R = \frac{u^2 \sin(2\theta)}{g}\)

R range (m)

u initial velocity (m/s)

θ launch angle (°)

g 9.8 m/s²

Horizontal distance traveled. Maximum at θ = 45° for level ground.

Pro tip: Range is same for complementary angles (30° and 60° give same R).

Initial speed u (m/s)

Launch angle θ (degrees)

g (m/s²)

Range R (m)

—

u = 25 m/s, θ = 45°. R = 625 × sin90° / 9.8 = 63.8 m

🔄

Rotational Motion

4 formulas · KCSE & CBC

Angular Speed

KCSE 2018,2020,2022

\(\omega = \frac{\Delta \theta}{\Delta t}\)

ω angular speed (rad/s)

Δθ angular displacement (rad)

Δt time (s)

Rate of change of angular position.

Pro tip: Convert rpm to rad/s: multiply by 2π/60. 1 rpm = 0.1047 rad/s.

Angular displacement Δθ (rad)

Time Δt (s)

Angular Speed ω (rad/s)

—

Wheel rotates 10π radians in 5 seconds. ω = 10π/5 = 6.28 rad/s

Angular Acceleration

KCSE 2019,2021

\(\alpha = \frac{\Delta \omega}{\Delta t}\)

α angular acceleration (rad/s²)

Δω change in angular velocity (rad/s)

Δt time (s)

Rate of change of angular velocity.

Pro tip: Analogous to linear acceleration a. τ = Iα is Newton's Second Law for rotation.

Change in ω Δω (rad/s)

Time Δt (s)

Angular Acceleration α (rad/s²)

—

Flywheel speeds from 2 rad/s to 8 rad/s in 3 s. α = (8-2)/3 = 2 rad/s²

Tangential Velocity

KCSE 2017,2020,2022

\(v = r\omega\)

v tangential speed (m/s)

r radius (m)

ω angular speed (rad/s)

Linear speed of a point on a rotating object.

Pro tip: At the center (r=0), v=0. At the edge, v is maximum.

Radius r (m)

Angular speed ω (rad/s)

Tangential Velocity v (m/s)

—

Wheel radius 0.4 m, ω = 15 rad/s. v = 0.4 × 15 = 6 m/s

Torque

KCSE 2018,2020,2022

\(\tau = rF\sin\theta\)

τ torque (N·m)

r lever arm (m)

F force (N)

θ angle between r and F

Rotational equivalent of force. Maximum torque when force is perpendicular (θ=90°).

Pro tip: Wrench works best when force is perpendicular to handle — sin90° = 1.

Lever arm r (m)

Force F (N)

Angle θ (degrees)

Torque τ (N·m)

—

Force 50 N at 0.3 m, angle 90°. τ = 0.3 × 50 × 1 = 15 N·m

🧪

Properties of Matter

4 formulas · KCSE & CBC

Shear Modulus

KCSE 2019,2021

\(G = \frac{\text{shear stress}}{\text{shear strain}}\)

G shear modulus (Pa)

F force (N)

A area (m²)

Δx lateral displacement (m)

h height (m)

Measures material's resistance to shape change under shear stress.

Pro tip: Rubber has low G (deforms easily), steel has high G (stiff).

Force F (N)

Area A (m²)

Displacement Δx (m)

Height h (m)

Shear Modulus G (Pa)

—

F=1000 N, A=0.01 m², Δx=0.002 m, h=0.1 m. G = 5×10⁶ Pa

Surface Charge Density

KCSE 2018,2020,2022

\(\sigma = \frac{Q}{A}\)

σ surface charge density (C/m²)

Q charge (C)

A area (m²)

Charge per unit area on a surface. For conductors, charge concentrates at sharp points.

Pro tip: Lightning rods work because high σ at sharp points causes corona discharge.

Charge Q (C)

Area A (m²)

Surface Charge Density σ (C/m²)

—

Q = 5×10⁻⁶ C, area = 0.1 m². σ = 5×10⁻⁵ C/m²

Charge Density

KCSE 2019,2021

\(\rho = \frac{Q}{V}\)

ρ volume charge density (C/m³)

Q charge (C)

V volume (m³)

Charge per unit volume. Used for continuous charge distributions.

Pro tip: For a uniformly charged sphere, total Q = ρ × (4/3)πR³.

Charge Q (C)

Volume V (m³)

Volume Charge Density ρ (C/m³)

—

Q = 2×10⁻⁸ C, V = 1×10⁻⁶ m³. ρ = 0.02 C/m³

Energy Density

KCSE 2020,2022

\(u = \frac{\text{Energy}}{\text{Volume}}\)

u energy density (J/m³)

Energy (J)

Volume (m³)

Energy stored per unit volume.

Pro tip: Energy density in a capacitor: u = ½ε₀E².

Energy (J)

Volume (m³)

Energy Density u (J/m³)

—

Battery stores 5000 J in 0.02 m³. u = 250,000 J/m³

⚡

Electricity

9 formulas · KCSE & CBC

Ohm's Law

KCSE 2017, 2019, 2020, 2022

\(V = IR\)

V Voltage (V)

I Current (A)

R Resistance (Ω)

The most fundamental electrical equation. Voltage drives current through resistance.

Pro tip: Water analogy: V = water pressure, I = flow rate, R = pipe narrowness.

Current I (A)

Resistance R (Ω)

Voltage V (Volts)

—

A current of 3 A flows through 10 Ω resistor. V = IR = 3 × 10 = 30 V

Electrical Power

KCSE 2018, 2020, 2021

\(P = IV\)

P Power (W)

I Current (A)

V Voltage (V)

Rate of electrical energy consumed or generated.

Pro tip: Your electricity bill is in kWh. 1 kWh = 3,600,000 J.

Current I (A)

Voltage V (V)

Power P (Watts)

—

A device draws 2 A at 240 V. P = IV = 2 × 240 = 480 W

Electrical Energy

KCSE 2019, 2021

\(E = Pt\)

E Energy (J)

P Power (W)

t time (s)

Total energy consumed over time.

Pro tip: 1 kWh = 3.6 × 10⁶ J. Always convert hours to seconds for Joules.

Power (W)

Time (s)

Energy E (Joules)

—

A 500 W heater runs for 120 s. E = Pt = 500 × 120 = 60,000 J

Resistivity

KCSE 2018, 2022

\(R = \rho \frac{L}{A}\)

R Resistance (Ω)

ρ resistivity (Ω·m)

L length (m)

A cross-section area (m²)

Resistance depends on material, length, and thickness.

Pro tip: Copper ρ = 1.7×10⁻⁸ Ω·m. Nichrome ρ = 1.1×10⁻⁶.

Resistivity ρ (Ω·m)

Length L (m)

Area A (m²)

Resistance R (Ω)

—

ρ = 1.7×10⁻⁸, L = 2 m, A = 10⁻⁶ m². R = 0.034 Ω

Series Resistance

KCSE 2017, 2019, 2021

\(R_{total} = R_1 + R_2 + R_3\)

R_total total resistance (Ω)

In series, resistances add up. Same current flows through all.

Pro tip: Adding more resistors in series always increases total resistance.

R₁ (Ω)

R₂ (Ω)

R₃ (Ω, 0 if none)

Total Resistance (Ω)

—

R₁ = 4 Ω, R₂ = 6 Ω. R_total = 4 + 6 = 10 Ω

Parallel Resistance

KCSE 2018, 2020, 2022

\(\frac{1}{R_T} = \frac{1}{R_1} + \frac{1}{R_2}\)

R_T total resistance (Ω)

Parallel paths share current. Total resistance is always less than the smallest resistor.

Pro tip: For two equal resistors in parallel: R_total = R/2.

R₁ (Ω)

R₂ (Ω)

Total Resistance (Ω)

—

R₁ = 6 Ω, R₂ = 3 Ω. 1/R = 1/6 + 1/3 = 1/2 → R = 2 Ω

Magnetic Flux

KCSE 2019,2021,2022

\(\Phi = BA\cos\theta\)

Φ magnetic flux (Wb)

B magnetic field (T)

A area (m²)

θ angle

Magnetic field lines passing through a surface. Changing flux induces EMF.

Pro tip: 1 Weber = 1 T·m². Maximum flux when B ⊥ area (θ=0).

Magnetic field B (T)

Area A (m²)

Angle θ (degrees)

Magnetic Flux Φ (Wb)

—

B = 0.5 T, A = 0.2 m², θ = 30°. Φ = 0.0866 Wb

Induced Voltage

KCSE 2017,2020,2022

\(\varepsilon = -N\frac{\Delta\Phi}{\Delta t}\)

ε induced EMF (V)

N number of turns

ΔΦ change in flux (Wb)

Δt time (s)

Changing magnetic flux induces voltage (Faraday's Law).

Pro tip: Generators, transformers, and induction cooktops all rely on this.

Number of turns N

Change in flux ΔΦ (Wb)

Time Δt (s)

Induced EMF ε (V)

—

N=100, ΔΦ=0.04 Wb, Δt=0.02 s. ε = 100 × 0.04/0.02 = 200 V

Coulombs Law

KCSE 2018,2020,2021

\(F = k\frac{q_1 q_2}{r^2}\)

F electrostatic force (N)

k 8.99×10⁹ N·m²/C²

q₁,q₂ charges (C)

r distance (m)

Force between two point charges. Like charges repel, opposites attract.

Pro tip: k = 1/(4πε₀). The force is enormous at short range.

Charge q₁ (C)

Charge q₂ (C)

Distance r (m)

Force F (N)

—

q₁=2×10⁻⁶, q₂=3×10⁻⁶, r=0.1 m. F = 5.4 N

🌊

Waves and Sound

6 formulas · KCSE & CBC

Wave Speed

KCSE 2017, 2019, 2021, 2022

\(v = f\lambda\)

v speed (m/s)

f frequency (Hz)

λ wavelength (m)

All waves obey this relationship. Speed is fixed by medium; frequency and wavelength trade off.

Pro tip: Light in vacuum: v = 3×10⁸ m/s. Sound in air: ~340 m/s.

Frequency f (Hz)

Wavelength λ (m)

Wave speed v (m/s)

—

f = 200 Hz, λ = 1.7 m. v = fλ = 340 m/s

Frequency and Period

KCSE 2018, 2020

\(f = \frac{1}{T}\)

f frequency (Hz)

T period (s)

Frequency is how many complete waves per second. Period is the time for one complete wave.

Pro tip: Mains electricity in Kenya: f = 50 Hz, so T = 0.02 s.

Period T (s)

Frequency f (Hz)

—

T = 0.004 s. f = 1/T = 250 Hz

Doppler Effect

KCSE 2019, 2021

\(f' = f\frac{v}{v \pm v_s}\)

f' observed freq

f source freq

v wave speed

v_s source speed

Frequency appears higher when source approaches, lower when it recedes.

Pro tip: Use v - v_s when approaching (higher pitch), v + v_s when receding (lower pitch).

Ambulance at 30 m/s emits 500 Hz, v=340 m/s. f' = 500 × 340/(340-30) = 548 Hz

Pendulum Period

KCSE 2017,2019,2021

\(T = 2\pi\sqrt{\frac{L}{g}}\)

T period (s)

L length (m)

g 9.8 m/s²

Time for one complete swing. Independent of mass and amplitude (small angles).

Pro tip: A pendulum clock runs slower at high altitude (g slightly lower).

Length L (m)

g (m/s²)

Period T (s)

—

L = 2.5 m. T = 2π√(2.5/9.8) = 3.17 s

Wavelength to Frequency

KCSE 2018,2020,2022

\(f = \frac{v}{\lambda}\)

f frequency (Hz)

v wave speed (m/s)

λ wavelength (m)

Convert wavelength to frequency if wave speed is known.

Pro tip: For EM waves in vacuum: f = 3×10⁸ / λ.

Wave speed v (m/s)

Wavelength λ (m)

Frequency f (Hz)

—

λ = 0.68 m, v = 340 m/s. f = 500 Hz

Hubbles Law

KCSE 2020,2022

\(v = H_0 r\)

v recession velocity (km/s)

H₀ Hubble constant (~70 km/s/Mpc)

r distance (Mpc)

Galaxies move away faster the farther they are. Evidence for expanding universe.

Pro tip: 1 Mpc = 3.26 million light-years.

Hubble constant (km/s/Mpc)

Distance r (Mpc)

Recession velocity v (km/s)

—

Galaxy at 100 Mpc. v = 70 × 100 = 7000 km/s

🔥

Thermal Physics

7 formulas · KCSE & CBC

Heat Energy

KCSE 2017, 2019, 2020, 2022

\(Q = mc\Delta T\)

Q heat (J)

m mass (kg)

c specific heat capacity (J/kg·K)

ΔT temperature change (K)

Heat needed to raise temperature depends on mass, material, and temperature change.

Pro tip: Water: c = 4184 J/kg·K (very high — why oceans moderate climate).

Mass (kg)

Specific heat c (J/kg·K)

Temp change ΔT (K)

Heat Q (Joules)

—

2 kg water, 20°C to 70°C. Q = 2 × 4184 × 50 = 418,400 J

Latent Heat

KCSE 2018, 2021

\(Q = mL\)

Q heat (J)

m mass (kg)

L specific latent heat (J/kg)

Heat absorbed or released during a change of state — no temperature change occurs.

Pro tip: Latent heat of vaporisation of water: 2.26×10⁶ J/kg. Why sweating cools so effectively.

Mass (kg)

Latent heat L (J/kg)

Heat Q (Joules)

—

Melt 0.5 kg of ice. Q = 0.5 × 336000 = 168,000 J

Ideal Gas Law

KCSE 2017, 2020, 2022

\(PV = nRT\)

P pressure (Pa)

V volume (m³)

n moles

R 8.314 J/mol·K

T temperature (K)

Describes how pressure, volume, and temperature of an ideal gas are related.

Pro tip: ALWAYS convert to Kelvin: K = °C + 273.

Moles n

Temperature T (K)

Volume V (m³)

Pressure P (Pascals)

—

1 mol, 300 K, 0.025 m³. P = 99,768 Pa

Boyles Law

KCSE 2018, 2020, 2021

\(P_1 V_1 = P_2 V_2\)

P₁ initial pressure

V₁ initial volume

P₂ final pressure

V₂ final volume

At constant temperature, pressure and volume are inversely proportional.

Pro tip: Double the pressure → halve the volume.

P₁ (Pa)

V₁ (m³)

P₂ (Pa)

V₂ (m³)

—

P₁=100,000, V₁=0.4, P₂=200,000. V₂ = 0.2 m³

Heat Capacity

KCSE 2019,2021

\(C = \frac{Q}{\Delta T}\)

C heat capacity (J/K)

Q heat added (J)

ΔT temperature change (K)

Heat required to raise an object's temperature by 1 K.

Pro tip: Water has very high heat capacity. That's why hot water bottles stay warm.

Heat Q (J)

Temp change ΔT (K)

Heat Capacity C (J/K)

—

5000 J raises temp by 10 K. C = 500 J/K

Fahrenheit Conversion

KCSE 2017,2020

\(F = \frac{9}{5}C + 32\)

F Fahrenheit (°F)

C Celsius (°C)

Convert Celsius to Fahrenheit. Water freezes at 32°F, boils at 212°F.

Pro tip: -40° is same in both scales.

Celsius (°C)

Fahrenheit (°F)

—

37°C → F = 9/5×37 + 32 = 98.6°F

Heat of Fusion

KCSE 2018,2020,2022

\(Q = m L_f\)

Q heat (J)

m mass (kg)

L_f latent heat of fusion (J/kg)

Heat required to melt a solid without temperature change.

Pro tip: Water: L_f = 334,000 J/kg.

Mass (kg)

L_f (J/kg)

Heat Q (J)

—

Melt 2 kg ice. Q = 2 × 334,000 = 668,000 J

💡

Light and Optics

3 formulas · KCSE & CBC

Snells Law

KCSE 2017, 2019, 2021, 2022

\(n_1 \sin\theta_1 = n_2 \sin\theta_2\)

n₁, n₂ refractive indices

θ₁ angle of incidence

θ₂ angle of refraction

Light bends when entering a new medium. Higher refractive index = slower light = more bending.

Pro tip: Glass n ≈ 1.5, water n ≈ 1.33, diamond n = 2.42. Air n ≈ 1.

Light hits glass (n=1.5) at 30°. sinθ₂ = sin30°/1.5 = 0.333 → θ₂ = 19.5°

Refractive Index

KCSE 2018, 2020

\(n = \frac{c}{v}\)

n refractive index

c 3×10⁸ m/s

v speed in medium

How much slower light travels in a medium compared to vacuum. Always ≥ 1.

Pro tip: Diamond's high n (2.42) causes total internal reflection → sparkle.

Speed in medium (m/s)

Refractive index n

—

Light travels at 2×10⁸ m/s in glass. n = 3×10⁸ ÷ 2×10⁸ = 1.5

Lens Formula

KCSE 2017, 2019, 2022

\(\frac{1}{f} = \frac{1}{u} + \frac{1}{v}\)

f focal length (m)

u object distance (m)

v image distance (m)

Predicts where a lens forms an image.

Pro tip: Magnification M = v/u. Camera lenses, eyes, microscopes all use this.

Object distance u (m)

Image distance v (m)

Focal length f (m)

—

Object 0.3 m, image 0.6 m. 1/f = 1/0.3 + 1/0.6 = 5 → f = 0.2 m

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Modern Physics

4 formulas · KCSE & CBC

Mass-Energy Equivalence

KCSE 2019, 2021, 2022

\(E = mc^2\)

E energy (J)

m mass (kg)

c 3×10⁸ m/s

Mass and energy are interchangeable. The basis of nuclear power.

Pro tip: 1 kg of mass → 9×10¹⁶ J.

Mass (kg)

Energy E (Joules)

—

m = 0.001 kg. E = 0.001 × 9×10¹⁶ = 9×10¹³ J

Photon Energy

KCSE 2018, 2020, 2022

\(E = hf\)

E energy (J)

h 6.63×10⁻³⁴ J·s

f frequency (Hz)

Light comes in discrete packets called photons. Higher frequency = more energetic.

Pro tip: UV photons can break chemical bonds and damage DNA.

Frequency f (Hz)

Photon Energy E (Joules)

—

f = 6×10¹⁴ Hz. E = 6.63×10⁻³⁴ × 6×10¹⁴ = 3.98×10⁻¹⁹ J

Radioactive Decay

KCSE 2017, 2019, 2021

\(N = N_0 e^{-\lambda t}\)

N remaining atoms

N₀ initial atoms

λ decay constant

t time

Radioactive atoms decay exponentially. After each half-life, half the remaining atoms decay.

Pro tip: Half-life t₁/₂ = ln2/λ = 0.693/λ. Carbon-14 half-life = 5730 years.

N₀=1000, λ=0.1 s⁻¹, t=10 s. N = 1000 × e⁻¹ = 368 atoms

Cell Potential

KCSE 2019,2021

\(E^0_{cell} = E^0_{red} - E^0_{oxid}\)

E°_cell standard cell potential (V)

E°_red reduction potential

E°_oxid oxidation potential

The voltage produced by an electrochemical cell. Positive E° means spontaneous reaction.

Pro tip: Daniel cell: Zn/Zn²⁺ and Cu/Cu²⁺ gives ~1.1 V.

Zn: E°_oxid = 0.76 V, Cu: E°_red = 0.34 V. E°_cell = 0.34 - (-0.76) = 1.10 V

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