Kinematics

Vectors and Scalars

Vectors: quantities that are fully described by a magnitude and a direction (e.g. velocity, displacement, acceleration, force, etc.)

Scalars: quantities that are fully described by just a magnitude (e.g. speed, mass, temperature, distance, time, etc.)

Vector Arithmetic

    Multiplying a Vector by a Scalar:

  • If the scalar has a negative value, reverse the vector to point in the opposite direction
  • Dividing by a scalar is the same as multiplying by the reciprocal (e.g. a/2 is the same as ½ x A)

    Adding Vectors (Resultant Vectors):

  • When vectors have the same direction, add the magnitudes
  • When vectors have opposite directions, subtract the magnitudes

    Resolving a Vector:

  • Reverse the process of adding two perpendicular vectors
  • Use trigonometry laws (SohCahToa) to break up vectors into x and y components

Motion

    Motion Quantities

  • Position (s): where an object is relative to a reference point (vector)
  • Displacement: defined as the change in position, where displacement = final position - initial position (vector)
  • Distance: length of the path followed by an object (scalar)
  • Velocity (v): defined as the rate of change of position, hence v = Δs/Δt (vector)
  • Speed: defined as distance divided by time, or d/t (scalar)
  • Acceleration (a): defined as the rate of change of velocity, hence a = Δv/Δt (vector)

    Displacement vs Time Graphs

  • The gradient of lines represents the velocity

    Velocity vs Time Graphs

  • The gradient of lines represents the acceleration
  • The area under lines/curves represents the displacement of the body:
    • Above the x-axis represents a positive displacement
    • Below the x-axis represents a negative displacement

    Free Fall

  • When an object is dropped near the surface of the Earth
  • Only gravity acts on objects
  • Objects are uniformly accelerating, having an acceleration pointing downards with a magnitude of: g = -9.81 ms-2

Equations of Motion

  • These equations connect motion quantities when the acceleration is constant, and are the fundamentals of kinematics

    • v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time
    • s = (u+v)/2, where s is the displacement, v is the final velocity, u is the initial velocity, and t is the time
    • s = ut + ½at2, where s is the displacement, v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time
    • v2 = u2 + 2as, where s is the displacement, v is the final velocity, a is the acceleration, and t is the time

    Projectile Motion

    The motion of an object thrown or projected into the air.


      Solving Projectile Motion Problems

    • The horizontal and vertical components of objects can be treated and solved separately from each other
    • The only factor that connects the horizontal and vertical components is time
    • Convert velocities and angles into vertical and horizontal components of speed, so the equations of motions can be used

      Calculating the Maximum Height

    • Determined by the vertical quantities of motion
    • Use the vertical component of the initial speed, a = -9.81, and v = 0 (the vertical speed is 0 when the maximum height is reached)

      Calculating the Time of Flight

    • Determined by the vertical quantities of motion
    • Use the vertical component of the initial speed, and the velocity when the object hits the ground

      Calculating the Range (Horizontal Distance)

    • Determined by the horizontal quantities of motion
    • Use the horizontal component of the initial speed

    Fluid Resistance

    Fluid: a substance that does not have a fixed shape and can flow easily (e.g. air and water).


      Air Resistance (Projectile Motion)

    • The trajectory is a different shape (it is not parabolic)
    • The maximum height is lower, and the range is shorter
    • The velocity will be lower
    • An object travelling upwards will reach its maximum height sooner, so the time of flight when going up is less
    • An object travelling downwards will experience an upwards resistance, so the time of flight when going down is greater

      Air Resistance (Free Fall)

    • As an object falls its speed increases, and so does the air resistance
    • Eventually air resistance becomes equal to the force of gravity, and the object stops accelerating - thus reaching its terminal velocity