The maths you need

1. Basic Mathematical Concepts

Multiplication & Notation

• If there is no multiplication sign $(×)$ or dot $(∙)$, numbers or symbols next to each other imply multiplication.
  • Example: m₁m₂ means $m₁ × m₂.$
  • Alternative notations: $m₁ ∙ m₂$ or $m₁ × m₂.$

Decimals & Commas

• Commas in numbers indicate decimal points in scientific notation.
  • Example: 4.5 means 4,5 in some calculators.
• Avoid confusion when multiplying:
  • 4,5 × 4,5 is 20.25, not 4,5 + 4,5.

Scientific Notation & Exponents

• Divisions often use exponents:
  • Example: $0.5 ms = 5 × 10⁻⁴ s.$
• Avoid common mistakes:
  • $0.5 ms ≠ 0.0005 s$ (correct value: $0.0005 s = 5 × 10⁻⁴ s$).

Variables & Symbols

• A variable represents a changing quantity.
• Scientific variables use single-letter symbols:
  • $v$= velocity, $a$ = acceleration, $F$ = force.
• Subscripts provide extra meaning:
  • $v₁$ (initial velocity), v₂ (final velocity).
• Capital vs lowercase letters matter:
  • $V$ = voltage, $v$= velocity.
  • R = resistance, $P$ = pressure.

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2. Solving for an Unknown

To determine an unknown value in an equation, isolate the required variable by applying basic mathematical operations:

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3. Step-by-Step Problem-Solving

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Key Takeaways

✔ Use basic algebra to isolate variables. ✔ Apply scientific notation for very small or large numbers. ✔ Recognise common physics equations and their rearrangements. ✔ Break problems into step-by-step solutions for clarity. ✔ Check units and notation for accuracy.

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3. Statistics

Key Definitions

• Dependent variable: The result or effect of an experiment.
• Independent variable: The cause or input that affects the outcome.
• Control variable: A variable that remains constant to ensure a fair test.

Understanding Correlation

• Correlation means two variables appear related but do not necessarily cause each other.
• A change in one variable might be due to an external factor.
• Graphs help us analyse these relationships.

Measures of Central Tendency

• Mean: The average of a data set.
  • Example: In the series $1, 3, 5, 7, 9:$
  • $Mean = (1 + 3 + 5 + 7 + 9) / 5 = 5$
• Median: The middle value when data is arranged in order.
  • Example: In the series $1, 3, 5, 7, 9:$
  • $Median = 5$
• Mode: The most frequently occurring value in a data set.
  • Example: In the series $1, 2, 2, 3, 3, 3, 4, 5:$
  • $Mode = 3$

Proportionality in Science

• In science, proportionality helps establish relationships between variables.
• Example: $Momentum (p) = mass (m) \times velocity (v)$
  • Momentum is directly proportional to both mass and velocity.
  • If either increases, so does momentum.

Constants in Equations

• Some formulas include constants, which are fixed values used to relate variables.
• Examples:
  • Gravitational constant $(G)$ in Newton's Law of Universal Gravitation.
  • Proportionality constant $(k)$ in various equations.

Summary

• Statistics in physics help us understand relationships between variables.
• Key concepts include correlation, mean, median, mode, and proportionality.
• Constants are used in equations to standardise relationships.

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4. Area of a Triangle

The area (AA) of a triangle is calculated as:

$A = \frac{1}{2} \times \text{Base} \times \text{Height}$

• Example: A triangle with $base = 5 cm$ and $height = 3 cm$ has an area of: $A = \frac{1}{2} \times 5 \times 3 = :highlight[7.5 \text{ cm}^2]$

• This is useful for graphs of motion and acceleration.

Pythagoras' Theorem – Finding Triangle Sides

In a right-angled triangle, the hypotenuse $(cc)$ is related to the other two sides:

$c^2 = h^2 + b^2$

where:

• $hh = height$
• $bb = base$
• $cc = hypotenuse$

Why This is Important in Physics

• Vector Addition: When dealing with forces, knowing triangle side lengths helps determine resultant force strength and direction.
• Graph Interpretation: Used when analysing motion and areas under curves in kinematics.

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5. Trigonometry in Physics

• Trigonometry is used to calculate unknown sides or angles in right-angled triangles.
• The three main trigonometric ratios:
  • $sin θ$ $= opposite / hypotenuse$
  • $cos θ = adjacent / hypotenuse$
  • $tan θ = opposite / adjacent$
• A useful mnemonic: SOHCAHTOA
  • $SOH → Sine = Opposite/Hypotenuse$
  • $CAH → Cosine = Adjacent/Hypotenuse$
  • $TOA → Tangent = Opposite/Adjacent$

Key Definitions

• Hypotenuse: The longest side in a right-angled triangle, always opposite the right angle.
• Opposite Side: The side opposite to the given angle.
• Adjacent Side: The side next to the given angle (but not the hypotenuse).

Inverse Trigonometric Functions

• Sometimes, you need to find the angle when two sides are known.
• Use inverse functions:
  • $sin⁻¹ (opposite/hypotenuse)$
  • $cos⁻¹ (adjacent/hypotenuse)$
  • $tan⁻¹ (opposite/adjacent)$
• Example: If opposite = 4 cm and hypotenuse = 5 cm, then: $θ = sin⁻¹(4/5) = :highlight[53.1°]$

Summary of Key Concepts

• Use SOHCAHTOA to solve right-angled triangles.
• Inverse trigonometry helps find unknown angles.
• Angles and distances in space can be calculated using trigonometry.
• Always check that your calculator is in degree mode when solving.

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6. Understanding Graphs in Physics

• Graphs are essential for interpreting motion, forces, and chemical reactions.
• Cartesian Coordinates help locate points using:
  • x-coordinate: Left-right position.
  • y-coordinate: Up-down position.

2. Working with Coordinates

• A point on a graph is written as an ordered pair: $(x, y).$
• Positive values:
  • Right of the $y-axis$ $(x > 0).$
  • Above the $x-axis$ $(y > 0).$
• Negative values:
  • Left of the $y-axis$ $(x < 0).$
  • Below the $x-axis$ $(y < 0).$

3. Identifying Linear Relationships

• A straight-line graph represents a direct relationship.
• The general equation of a straight line:

where: $y=mx+c$

• $m = gradient$ $(slope).$
• c = y-$$intercept.

4. Gradient (Slope) of a Graph

• $Gradient (m)$ shows how steep the line is.

• Formula: $m = \frac{\text{change in } y}{\text{change in } x} = \frac{y_2 - y_1}{x_2 - x_1}$

• Interpreting the gradient:

  • Positive gradient: Line slopes upward (y increases as x increases).
  • Negative gradient: Line slopes downward (y decreases as x increases).
  • Zero gradient: Horizontal line (constant value).

5. Common Graphs in Physical Sciences

• Directly Proportional Graph:

  • Passes through the origin $(0,0).$
  • Equation: , where $k$ is a constant. $y = kx$

• Inverse Proportion Graph:

  • Equation: . $y = \frac{k}{x}$

  • As x increases, y decreases (curved graph).

• Quadratic Graph:

  • Equation: . $y = ax^2 + bx + c$

  • Produces a parabola.

6. Application in Physics

• Velocity-Time Graph:
  • Gradient = acceleration.
  • Area under the graph = displacement.
• Position-Time Graph:
  • Gradient = velocity.
• Force vs. Acceleration Graph:
  • Gradient = mass of the object (from Newton's Second Law: ). $F = ma$

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7. Understanding Circles in Physics

• A circle is a two-dimensional shape with a curved boundary.
• Important parts of a circle:

  • $Diameter$ $(d)$: The widest part of a circle, passing through the centre.

  • $Radius$ $(r)$: Half of the diameter. Formula: . $r = \frac{d}{2}$

  • $Circumference$ $(C)$: The perimeter (outer boundary) of the circle. Formula: $C = 2\pi r$

  • $Area$ $(A)$: The space inside a circle. Formula: $A = \pi r^2$

2. Key Concepts for Physics

• Understanding "circumference":
  • Comes from the Latin word meaning "to carry in a circle".
  • Think of how planets orbit the Sun, or how the Earth moves around its axis.
• Applications in Physical Sciences:
  • Used in calculations for circular motion (e.g. planetary orbits, wheels, and gears).
  • Important for formulas involving rotational motion and wave properties.
  • Helps in understanding the relationship between distance, speed, and time in circular motion.

3. Using Circle Formulas in Exams

• You may need to calculate radius, diameter, circumference, or area using the formulas provided.
• Tip: These formulas are usually given in the exam, so focus on knowing how to apply them rather than memorising.