Kinematics Toolkit (DP IB Maths: AI HL)

What is kinematics?

• Kinematics is the branch of mathematics that models and analyses the motion of objects
• Common words such as distance, speed and acceleration are used in kinematics but are used according to their technical definition

What definitions do I need to be aware of?

• Firstly, only motion of an object in a straight line is considered
  • this could be a horizontal straight line
    • the positive direction would be to the right
  • or this could be a vertical straight line
    • the positive direction would be upwards

Particle

• A particle is the general term for an object
  • some questions may use a specific object such as a car or a ball

Time  seconds

• Displacement, velocity and acceleration are all functions of time
• Initially time is zero 

Displacement  m

• The displacement of a particle is its distance relative to a fixed point
  • the fixed point is often (but not always) the particle’s initial position
• Displacement will be zero  if the object is at or has returned to its initial position
• Displacement will be negative if its position relative to the fixed point is in the negative direction (left or down)

Distance  m

• Use of the word distance needs to be considered carefully and could refer to
  • the distance travelled by a particle
  • the (straight line) distance the particle is from a particular point
• Be careful not to confuse displacement with distance
  • if a bus route starts and ends at a bus depot, when the bus has returned to the depot, its displacement will be zero but the distance the bus has travelled will be the length of the route
• Distance is always positive

Velocity m s^-1

• The velocity of a particle is the rate of change of its displacement at time
• Velocity will be negative if the particle is moving in the negative direction
• A velocity of zero means the particle is stationary 

Speed  m s^-1

• Speed is the magnitude (a.k.a. absolute value or modulus) of velocity
  • as the particle is moving in a straight line, speed is the velocity ignoring the direction
    • if
    • if

Acceleration  m s^-2

• The acceleration of a particle is the rate of change of its velocity at time
• Acceleration can be negative but this alone cannot fully describe the particle’s motion
  • if velocity and acceleration have the same sign the particle is accelerating (speeding up)
  • if velocity and acceleration have different signs then the particle is decelerating (slowing down)
  • if acceleration is zero the particle is moving with constant velocity 
  • in all cases the direction of motion is determined by the sign of velocity

Are there any other words or phrases in kinematics I should know?

• Certain words and phrases can imply values or directions in kinematics
  • a particle described as “at rest” means that its velocity is zero,
  • a particle described as moving “due east” or “right” or would be moving in the positive horizontal direction
    • this also means that
  • a particle “dropped from the top of a cliff” or “down” would be moving in the negative vertical direction
    • this also means that

What are the key features of a velocity-time graph?

• The gradient of the graph equals the acceleration  of an object
• A straight line shows that the object is accelerating at a constant rate
• A horizontal line shows that the object is moving at a constant velocity
• The area between graph and the x-axis tells us the change in displacement of the object
  • Graph above the x-axis means the object is moving forwards
  • Graph below the x-axis means the object is moving backwards

• The total displacement of the object from its starting point is the sum of the areas above the x-axis minus the sum of the areas below the x-axis
• The total distance travelled by the object is the sum of all the areas
• If the graph touches the x-axis then the object is stationary at that time
• If the graph is above the x-axis then the object has positive velocity and is travelling forwards
• If the graph is below the x-axis then the object has negative velocity and is travelling backwards