SPH4U - Unit 1 - Kinematics

Position, Velocity, and Acceleration

• Position - The location of an object at a particular time

• Displacement - The change in position of an object from its initial position

  • It indicates the final distance and direction moved from a starting position

  • It does not necessarily equal the total distance traveled

• Average Velocity - Displacement divided by the time interval over which displacement occurs

• Instantaneous Velocity - Velocity at a specific instant, or the average velocity over an infinitesimal time interval

• Position-time graphs compare the position of an object, relative to a specific point, to the amount of time passed

  • The vertical axis indicates the position

  • The slope indicates the speed

    • A steeper slope will be moving faster

  • A decreasing slope indicates that the object is moving towards a specific reference point

  • An increasing slope indicates that the object is moving away from a specific reference point

  • The instantaneous speed can be calculated by taking the slope of the tangent line at a specific point on the graph

    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

• Velocity-time graphs compare the velocity of an object to the amount of time passed

  • The vertical axis indicates the velocity

  • If the graph is above the x-axis, it is moving away from a specific point

  • If the graph is below the x-axis, it is moving towards a specific point

    • If moving away from the specific point in one direction is positive, going the opposite direction (towards the point) makes the displacement negative, so the total velocity becomes negative

  • The area between the graph and the horizontal axis shows how far an object travels, but it cannot determine the location of the object

  • The slope indicates the acceleration

    • A horizontal line means that it is travelling at a uniform velocity

    • The specific location of the object cannot be determined

• Acceleration - How much an object’s speed changes in one second, measured in m/s²

  • An increasing velocity means acceleration is in the same direction as the motion

  • A decreasing velocity means acceleration is in the opposite direction of motion

  • An object in free fall has an acceleration of 10m/s²

• Acceleration-time graphs compare the acceleration of an object over an interval of time

  • 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

• If an object is moving with uniform acceleration, five equations can be used to model the relationship between the initial velocity (v_o), the final velocity (v_o), the displacement (∆d or ∆x), acceleration (a), and time elapsed (t)

  • Always declare the positive and negative directions

  • Use words to indicate the portion of motion being considered and solved for

    ∆d = [(v2 + v1)/2](∆t)

    v2 = v1 + a(∆t)

    ∆d = v1(∆t) + (a/2)(∆t)²

    v2² = v1² + 2a(∆d)

    ∆d = v2(∆t) - (a/2)(∆t)²

Projectile Motion

• The vertical motion has an acceleration according to gravity, and can be represented by the five kinematic equations

• The horizontal motion has an acceleration of zero, and can be represented by v=d/t

• 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

• When given a velocity, if it is not entirely horizontal or vertical, the x and y components of the velocity vector must be solved for

  • 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 the tangent to solve for the angle direction

• Equilibrium - When the forces on an object are balanced so that it moves in either a straight line at a constant speed, or remains at rest

  • Acceleration is in the direction in which forces are unbalanced

  • The net force is in the direction in which the forces are unbalanced

Frames of Reference and Relative Motion

• Frame of Reference - A coordinate system relative to which motion is described or observed

• Relative Velocity - The velocity of an object relative to a specific frame of reference

  • The vector symbol v denotes velocity, followed by two capital letter subscripts to indicate relativity

  • The first subscript represents the moving object

  • The second subscript represents the frame of reference that the moving object is relative to