SPH3U - Unit 1 - Kinematics
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 = →dfinal - dinitial
Ex. If df =500m [East], and di = 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
Equation | Variable not in Equation |
|---|---|
∆→d = [(v2 + v1)/2](∆t) | a |
v2 = v1 + a(∆t) | ∆→d |
∆→d = v1(∆t) + (a/2)(∆t)² | v2 |
v2² = v1² + 2a(∆→d) | ∆t |
∆→d = v2(∆t) - (a/2)(∆t)² | v1 |
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