Equation ~ Vocabulary ~ Principle/Law

Math Skills

• Slope - m = ∆y/∆x (or rise/run)

• Pythagorean Theorem - a² + b² = c²

• Trigonometric Ratios -

  Sin = Opposite/Hypotenuse	Cos = Adjacent/Hypotenuse	Tan = Opposite/Adjacent
  Sine Law sin A/a = sin B/b = sin C/c	Cosine Law c² = a² + b² - 2abcosC

• Converting Units - multiply/divide by amount of current units in desired unit

  • 1 hour = 3600 seconds

  • 1 kilometre - 100 000 centimetres

Distance and Position

• Scalar - A measured quantity with its unit

  • Ex. Distance = 5 metres

  • Distance - The total length of the path travelled by an object

• Vector - A measured quantity with its unit and direction

  • Ex. Velocity = Speed and direction

  • Position - The location of an object based on a reference point

  • Displacement - Distance with a direction; A change in position

    • Can solve be solved three ways

    • Vector subtraction - ∆→d = →d_final - d_initial

      • Ex. If d_f =500m [East], and d_i = 0m, ∆→d = 500m - 0m = 500m [East]

    • Vector addition - ∆→d = ∆→d₁ + ∆→d₂

      • Declare + and - signs to represent directions

      • Ex. If ∆→d₁ = +10m, and ∆→d₂ = -2m, then ∆→d = 10m + (-2m) = +8m

    • Drawing Arrows

      • Declare a scale

      • Draw a measured arrow in the first direction

      • Continue drawing arrows as necessary, following measures and directions

      • Draw an arrow across the displacement and convert to find the displacement

      • Label arrows with drawn units, actual units, and directions

Speed and Velocity

• Speed (or average) - How quickly the object gets from one point to another

  • Speed = Total distance/Total time

    • Ex. 10km/h, 2m/s

• Velocity - The vector form of speed

  • Velocity = Total displacement/Total time

  • v = ∆→d/∆t

  • Ex. 7m [East]/s

LAB: Measurement, Precision, and Accuracy

• Accuracy - How close a measurement is to the correct value for that measurement

  • % error = (accepted - experimental)/accepted x (100%)

• Precision - The difference between a group of repeated measurements, also known as the spread

  • Precision = average - obtained

• Volume - The amount of space that a substance or object occupies

  • Displacement Method - Measuring volume by placing an object into a container with a known volume of water

• Significant Figures - All nonzero digits, as well as zeros after a decimal or between two nonzero numbers

• Density = mass/volume

Acceleration

• Acceleration - How quickly an object’s velocity changes over time

• Instantaneous Acceleration - The rate of change of velocity at a specific instance of time

• Acceleration due to gravity: a = 9.80m/s²

  • How objects fall towards earth without air resistance

• The object that is slower to reach the determined speed has a smaller acceleration

• Acceleration is a vector, so it will always have direction

• Acceleration will always be in distance/time²

Graphs of Motion

• Uniform Velocity - Motion at a constant speed in a straight line

• Non-Uniform Velocity - Motion that is not at a constant speed or not in a straight line

• Position-Time Graph - A graph describing the motion of an object, with position on the vertical axis and time on the horizontal axis

  • Slope of graph gives the velocity, and a steeper slope represents a greater velocity

  • Slopes and intercepts can be used to make an equation

  • A straight line will give a constant velocity

    Position-Time Graph Shape	Type of Motion
    Straight, horizontal line above the x-axis	At rest at a constant positive position Velocity = 0
    Straight, horizontal line below the x-axis	At rest at a constant negative position Velocity = 0
    Straight, upwards, diagonal line	Moving away from reference point Velocity = constant
    Straight, downwards, diagonal line	Moving towards reference point Velocity = constant
    Upward curve that becomes steeper	Accelerating away from reference point Velocity = not constant
    Downward curve that becomes steeper	Accelerating towards reference point Velocity = not constant

  • To find the slope of a straight line:

    • m = ∆y/∆x

  • To find the slope of a curved line:

    • Draw a tangent line that only touches the one point at the desired time

    • Find two points on the line and calculate slope

    • This is the velocity at that given moment

    • Instantaneous Velocity - The velocity of an object at a specific instant in time

• Velocity-Time Graph - A graph describing the motion of an object, with the velocity on the vertical axis and time on the horizontal axis

  • Velocity-time graphs for uniform velocity and uniform acceleration are always straight

  • Uniform Acceleration - Acceleration where the amount and direction is constant

  • Area of a velocity-time graph is equivalent to distance

  • Slope of a velocity-time graph is acceleration

    Velocity-Time Graph Shape	Type of Motion
    Horizontal line at x-axis	No motion, constant position Velocity = 0
    Horizontal line above x-axis	Moving away from reference point Velocity = constant
    Horizontal line below x-axis	Moving towards reference point Velocity = constant
    Diagonal upwards line	Uniform acceleration in positive direction
    Diagonal downwards line	Uniform acceleration in negative direction

• Acceleration-Time Graph - A graph describing motion of an object, with acceleration on the vertical axis and time on the horizontal axis

  • The area of an acceleration-time graph is velocity

    Acceleration-Time Graph Shape	Type of Motion
    Horizontal line at x-axis	No motion, constant position Velocity = 0
    Horizontal line above x-axis	Uniform acceleration in positive direction
    Horizontal line below x-axis	Uniform acceleration in negative direction

Key 5 Equations for Uniform Acceleration

• If different directions are present, use - and + to establish directions

LAB: Position and Speed of an Object

• In a distance/time graph, the slope and the velocity are equal

  • m = (d/t) = v

• A line of best fit can estimate the general path of an objet

Acceleration of Earth’s Gravity

• Falling objects have a downwards uniform accelerating motion

• Gravity on objects near earth’s surface has an acceleration of 9.8m/s² [Down]

• In a curve, velocity is 0m/s at the max

Solving 2D Motion

• Method 1 - Adding Two Vectors with a Scale Diagram

  • Pick a scale and state it

  • Draw vector 1 with an arrow, following cardinal directions

    • [(direction 1)(degrees)(direction 2)] means to start at direction 1 and turn the specified amount of degrees to direction 2

  • Draw vector 2 from the end of vector 1, following cardinal directions from that point

  • Finish the triangle from the start of vector 1 to vector 2

  • Measure with a ruler and protractor and convert to scale

  • The result of adding two vectors is the resultant vector

• Method 2 - Component Method

  • Find x and y components of each vector using trigonometry

    • Asinθ = y

    • Acosθ = x

  • Group x components and add (consider directions)

  • Group y components and add (consider directions)

  • Find the total using the Pythagorean Theorem

    • A = √[(total x)²+ (total y)²]

  • Use tangent to solve for direction

LAB: Acceleration

• g = a/sinθ

• Gravity is the slope of and acceleration/sinθ graph

Projectile Motion

• Time fall is independent of the horizontal velocity

• An increase in speed = increased range, but does not affect time

• An increase in height = increased range and time

• Mass does not change speed/time/range

• Projectile Motion - The motion of a projectile is guided by gravity only and assumes no air resistance

  • The path taken is in the form of a parabola

  • The motion can be broken down into horizontal and vertical components

  • The motion of the horizontal direction is a constant velocity

  • In the vertical direction, the object experiences acceleration due to gravity, so it is always 9.8m/s²

  • Horizontal and vertical parts behave differently, so there are separate equations

    • Horizontal - v = d/t

      • For range - R = (v₁²sin2θ/g)

    • Vertical - big 5 equations

    • Time is equivalent in both cases

LAB: Motion in 2D