Unit 01 - Grade 11-12 Physics

Foundations, Measurement, and Math

Build the skills every physics unit depends on: SI units, prefixes, scientific notation, significant figures, uncertainty, graphing, vectors, algebra, trigonometry, and clear problem-solving habits.

Lesson roadmap

What Students Should Master in This Unit

This first unit is not just a review. It is the toolkit for the entire physics course. Students who are strong with units, algebra, graphs, uncertainty, and vectors usually solve later topics faster and make fewer mistakes.

Measure clearly

Use SI units, prefixes, significant figures, precision, accuracy, and uncertainty correctly.

Calculate carefully

Rearrange formulas, use scientific notation, convert units, and keep answers reasonable.

Represent physics

Use graphs, vectors, components, diagrams, and written explanations to show thinking.

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Jump to a Topic

Physical Quantities SI Units and Prefixes Scientific Notation Significant Figures Measurement and Uncertainty Formula Skills Graphing Skills Vectors and Components Lab Skills Worked Examples Practice Problems Final Review
Core idea

1. Physical Quantities

A physical quantity is something that can be measured or calculated. Every complete physics answer needs both a number and a unit. For example, saying a car travels "20" is incomplete. Saying it travels "20 m" or moves at "20 m/s" gives meaning.

Scalar and vector comparison diagram A complete physics quantity needs value, unit, and sometimes direction Scalar 25 kg magnitude only Vector 12 m/s east magnitude plus direction
Scalars use a size only. Vectors must include both size and direction, which is why diagrams matter in physics.

Scalars and Vectors

Type Meaning Examples How to Write It
Scalar Has magnitude only. Mass, time, speed, distance, energy, temperature. 25 kg, 8 s, 12 m/s, 500 J.
Vector Has magnitude and direction. Displacement, velocity, acceleration, force, momentum. 25 m east, 12 m/s north, 9.8 m/s2 downward.
Student habit: Always ask, "Does direction matter?" If yes, the quantity is probably a vector.
Measurement language

2. SI Units and Prefixes

Physics uses the SI system because it keeps calculations consistent. Most formulas only work cleanly when values are converted into base SI units before substitution.

Metric prefix conversion diagram Metric prefixes are powers of ten kilo base centi milli micro 10^3 10^0 10^-2 10^-3 10^-6 moving right makes the number larger moving left makes the number smaller
Convert into SI base units before using physics formulas. Example: 35 cm = 0.35 m and 250 g = 0.250 kg.

Seven SI Base Units

Quantity SI Unit Symbol Common Physics Use
LengthmetermDistance, displacement, wavelength.
MasskilogramkgInertia, force, momentum, energy.
TimesecondsMotion, period, frequency, rates.
Electric currentampereACircuits and electromagnetism.
TemperaturekelvinKThermal physics and gases.
Amount of substancemolemolGas laws and chemistry connections.
Luminous intensitycandelacdLight measurement.

Important Derived Units

Force 1 N = 1 kg·m/s2 Used in Newton's laws.
Energy and Work 1 J = 1 N·m = 1 kg·m2/s2 Used in work, energy, and heat.
Power 1 W = 1 J/s Rate of energy transfer.
Pressure 1 Pa = 1 N/m2 Used in fluids and gases.
Charge 1 C = 1 A·s Used in electricity.
Frequency 1 Hz = 1 s-1 Cycles per second.

Metric Prefixes Students Must Know

Prefix Symbol Power of 10 Example
gigaG1091 GW = 1,000,000,000 W
megaM1061 MJ = 1,000,000 J
kilok1031 km = 1000 m
centic10-21 cm = 0.01 m
millim10-31 ms = 0.001 s
microµ10-61 µm = 0.000001 m
nanon10-91 nm = 0.000000001 m
Common mistake: Students often forget that the kilogram is already the SI base unit for mass. Convert grams to kilograms before using formulas like F = ma or p = mv.
Number sense

3. Scientific Notation

Scientific notation makes very large and very small quantities easier to read and calculate. A number in scientific notation has the form:

a × 10n, where 1 ≤ a < 10
Scientific notation diagram Move the decimal until one nonzero digit is left of it 45,000 m decimal moves 4 places left 4.5 x 10^4 m same value, cleaner form Large numbers use positive exponents; small decimals use negative exponents.
Scientific notation makes extreme physics quantities easier to compare, multiply, and divide.

Examples

  • 45,000 m = 4.5 × 104 m
  • 0.0032 s = 3.2 × 10-3 s
  • 6,370,000 m = 6.37 × 106 m

Operations with Powers of Ten

Multiplication (10a)(10b) = 10a+b Multiply coefficients and add exponents.
Division 10a / 10b = 10a-b Divide coefficients and subtract exponents.
Powers (10a)b = 10ab Multiply exponents.
Precision

4. Significant Figures

Significant figures show how precise a measurement is. They matter because physics answers should not look more precise than the measurements used to calculate them.

Significant figures ruler diagram Measurement precision depends on the tool scale 2.0 2.2 2.4 2.6 read certain digits, then estimate one more Example reading: about 2.35 cm
The last reported digit is usually estimated. Do not report more precision than the measuring tool supports.

Rules for Counting Significant Figures

  • All nonzero digits are significant: 352 has 3 significant figures.
  • Zeros between nonzero digits are significant: 3005 has 4 significant figures.
  • Leading zeros are not significant: 0.0042 has 2 significant figures.
  • Trailing zeros after a decimal are significant: 2.500 has 4 significant figures.
  • Trailing zeros without a decimal can be ambiguous: 1500 may have 2, 3, or 4 significant figures unless written in scientific notation.

Calculation Rules

Operation Rule Example
Multiplication and division Answer has the same number of significant figures as the least precise value. 2.4 × 3.18 = 7.632, report as 7.6
Addition and subtraction Answer has the same number of decimal places as the value with the fewest decimal places. 12.35 + 1.2 = 13.55, report as 13.6
Best habit: Keep extra digits during the calculation, then round only the final answer.
Lab readiness

5. Measurement, Accuracy, Precision, and Uncertainty

Accuracy and precision diagram Accuracy and precision are not the same thing accurate and precise precise but not accurate
Accuracy means close to the accepted value. Precision means repeated measurements are close to each other.

Accuracy vs. Precision

  • Accuracy means closeness to the accepted or true value.
  • Precision means repeatability or closeness of repeated measurements to each other.
  • A result can be precise but not accurate if a tool has a systematic error.

Types of Error

Error Type Meaning Example How to Reduce It
Random error Unpredictable variation between trials. Reaction time when using a stopwatch. Repeat trials and average results.
Systematic error A consistent bias in one direction. A scale reads 0.05 kg when empty. Calibrate equipment and correct zero errors.

Uncertainty Formulas

Absolute uncertainty measured value ± uncertainty Example: L = 2.35 ± 0.01 m.
Percent uncertainty % uncertainty = (absolute uncertainty / measured value) × 100% Use to compare precision between measurements.
Percent error % error = |experimental - accepted| / accepted × 100% Use when an accepted value is known.

Basic Uncertainty Propagation

  • For addition or subtraction, add absolute uncertainties.
  • For multiplication or division, add percent uncertainties.
  • For powers, multiply the percent uncertainty by the power.
Math toolkit

6. Formula Skills and Algebra

A formula is a relationship between quantities. Students should understand what each symbol means, what units it uses, and how to solve for any variable.

Formula rearranging diagram Use opposite operations to isolate the unknown F = ma solve for a F / m = a divide both sides by m Whatever you do to one side of an equation, do to the other side too.
Rearranging formulas is usually about undoing operations in reverse order while keeping the equation balanced.

Core Algebra Rules

Speed v = d / t Rearrange to d = vt or t = d / v.
Density ρ = m / V Rearrange to m = ρV or V = m / ρ.
Newton's second law F = ma Rearrange to m = F / a or a = F / m.
Slope m = (y2 - y1) / (x2 - x1) Slope often represents a physical quantity.
Linear equation y = mx + b m is slope, b is y-intercept.
Pythagorean theorem c2 = a2 + b2 Used for perpendicular vector components.

Important Math Formula Reference for Physics

Mean or average average = sum of values / number of values Useful for repeated trials and lab data.
Percent change % change = (new - old) / old × 100% Use signs carefully when values decrease.
Direct proportion y ∝ x means y = kx A straight line through the origin.
Inverse proportion y ∝ 1/x means xy = k As one variable increases, the other decreases.
Inverse-square relation y ∝ 1/r2 Used in gravity, electric fields, and light intensity.
Quadratic formula x = [-b ± √(b2 - 4ac)] / 2a Use when ax2 + bx + c = 0.
Rectangle area A = bh Often used for area under constant sections of graphs.
Triangle area A = 1/2 bh Often used for area under velocity-time graphs.
Circle formulas C = 2πr, A = πr2 Used in circular motion, waves, fluids, and fields.

Problem-Solving Method

  1. Read the question and identify what is being asked.
  2. List known values with units.
  3. Convert values into SI units when needed.
  4. Draw a diagram if the situation involves direction, motion, or forces.
  5. Choose the formula that connects the knowns to the unknown.
  6. Substitute numbers with units.
  7. Solve and check whether the answer is reasonable.
Data skills

7. Graphing Skills in Physics

Graphs show relationships. A graph can reveal patterns faster than a table of numbers. In physics, slope and area often have real meaning.

Graph slope and area diagram On physics graphs, slope and area often have units and meaning slope = rise / run area under graph x variable y variable
For example, slope on a position-time graph is velocity, while area under a velocity-time graph is displacement.

Graphing Checklist

  • Put the independent variable on the x-axis.
  • Put the dependent variable on the y-axis.
  • Label each axis with quantity and unit, such as time (s).
  • Choose an even scale that uses most of the graph space.
  • Plot points carefully and draw a best-fit line or curve.
  • Use two points on the best-fit line, not necessarily two data points, to calculate slope.

Key Graph Meanings

Graph Slope Means Area Means Common Unit
Position vs. timeVelocityUsually not used in Grade 11-12 physicsm/s
Velocity vs. timeAccelerationDisplacementm/s2 and m
Acceleration vs. timeChange in acceleration rateChange in velocitym/s
Force vs. displacementSpring constant if force is proportional to displacementWorkN/m and J
Voltage vs. currentResistanceUsually not usedohms

Linearization

Some relationships are curved. Linearization means changing the graph so the relationship becomes a straight line. For example, if d is proportional to t2, graph d vs. t2 instead of d vs. t.

Direction matters

8. Vectors and Components

Vectors are used throughout motion, forces, momentum, fields, and waves. A vector has magnitude and direction. Components split a vector into perpendicular parts, usually x and y.

Vector components diagram A vector can be split into perpendicular components A_x = A cos(theta) A_y = A sin(theta) A theta
If the angle is measured from the x-axis, the x-component uses cosine and the y-component uses sine.

Trigonometry Review

Sine sin(θ) = opposite / hypotenuse Useful for the component opposite the angle.
Cosine cos(θ) = adjacent / hypotenuse Useful for the component next to the angle.
Tangent tan(θ) = opposite / adjacent Useful for finding direction angles.

Vector Component Formulas

x-component Ax = A cos(θ) Use when θ is measured from the x-axis.
y-component Ay = A sin(θ) Use when θ is measured from the x-axis.
Magnitude from components A = √(Ax2 + Ay2) Use the Pythagorean theorem.
Direction from components θ = tan-1(Ay / Ax) Check the quadrant before final direction.
Resultant x-component Rx = Ax + Bx + Cx + ... Add signed components.
Resultant y-component Ry = Ay + By + Cy + ... Then find magnitude and direction.
Common mistake: If the angle is measured from the vertical axis instead of the horizontal axis, the sine and cosine roles may switch. Always draw the triangle.
Scientific communication

9. Lab Skills and Written Explanations

Strong physics students do more than calculate. They explain methods, support claims with evidence, and connect results to physical principles.

Lab Report Essentials

  • Purpose: State what relationship or principle is being tested.
  • Variables: Identify independent, dependent, and controlled variables.
  • Procedure: Explain enough steps that someone else could repeat the investigation.
  • Data table: Include headings, units, and repeated trials when appropriate.
  • Graph: Use correct labels, scale, best-fit line, and slope calculation.
  • Analysis: Show calculations and explain what the results mean.
  • Conclusion: Answer the purpose, use evidence, and discuss uncertainty or error.

Claim-Evidence-Reasoning

Claim A clear answer to the lab question.
Evidence Specific data, graph features, slopes, or calculated values.
Reasoning The physics principle that explains why the evidence supports the claim.
Worked examples

10. Worked Examples

Example 1: Unit conversion

A student runs 2.50 km in 12.0 min. What is the average speed in m/s?

Convert distance: 2.50 km = 2500 m.

Convert time: 12.0 min = 720 s.

v = d / t = 2500 m / 720 s = 3.47 m/s.

Example 2: Percent error

A lab group measures g = 9.52 m/s2. The accepted value is 9.80 m/s2. Find percent error.

% error = |9.52 - 9.80| / 9.80 × 100%

% error = 0.28 / 9.80 × 100% = 2.86%

Example 3: Slope of a graph

A position-time graph has points (2.0 s, 5.0 m) and (8.0 s, 23.0 m). Find the velocity.

Slope = (23.0 m - 5.0 m) / (8.0 s - 2.0 s)

Slope = 18.0 m / 6.0 s = 3.0 m/s.

On a position-time graph, slope represents velocity.

Example 4: Vector components

A force of 40.0 N acts 30.0 degrees above the horizontal. Find its x and y components.

Fx = F cos(θ) = 40.0 cos(30.0) = 34.6 N

Fy = F sin(θ) = 40.0 sin(30.0) = 20.0 N

Independent practice

11. Practice Problems

Try these without looking first. Then open the answer check to compare your reasoning.

1. Convert 75.0 cm to meters.

Answer

75.0 cm = 0.750 m.

2. Convert 0.00450 kg to grams.

Answer

0.00450 kg = 4.50 g.

3. Write 0.0000825 m in scientific notation.

Answer

8.25 × 10-5 m.

4. How many significant figures are in 0.03040?

Answer

4 significant figures: 3, 0, 4, and the final 0.

5. Calculate 4.20 m / 2.0 s and report with correct significant figures.

Answer

2.1 m/s. The limiting value has 2 significant figures.

6. A measurement is 18.6 cm ± 0.2 cm. Find percent uncertainty.

Answer

(0.2 / 18.6) × 100% = 1.08%, about 1.1%.

7. A cart moves from 1.0 m to 9.0 m in 4.0 s. Find average velocity.

Answer

v = displacement / time = (9.0 - 1.0) m / 4.0 s = 2.0 m/s.

8. A vector has components Ax = 6.0 m and Ay = 8.0 m. Find its magnitude.

Answer

A = √(6.02 + 8.02) = 10.0 m.

9. A 25.0 N force acts at 40.0 degrees above the horizontal. Find Fx.

Answer

Fx = 25.0 cos(40.0) = 19.2 N.

10. On a velocity-time graph, what does the area under the graph represent?

Answer

Displacement.

11. On a position-time graph, what does a steeper slope mean?

Answer

A greater speed or velocity magnitude.

12. A student finds density using m = 240 g and V = 80.0 cm3. Find density.

Answer

ρ = m / V = 240 g / 80.0 cm3 = 3.00 g/cm3, depending on significant figure interpretation for 240.

Final review

12. What to Know Before Moving On

  • Convert units into SI before using formulas.
  • Track significant figures and round only at the end.
  • Write answers with units and direction when needed.
  • Know the difference between accuracy, precision, random error, and systematic error.
  • Use slope and area to interpret physics graphs.
  • Break vectors into x and y components using trigonometry.
  • Use a consistent problem-solving format: knowns, unknown, formula, substitution, answer, reasonableness check.
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