Kinematics Toolkit (Cambridge International IGCSE Further Maths)

Displacement, Velocity & Acceleration

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$t$ seconds

  • Displacement, velocity and acceleration are all functions of time$t$
  • Initially time is zero $t=0$

• Displacement $s$ 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$s=0$ 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$d$ 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$v$m s^-1

  • The velocity of a particle is the rate of change of its displacement at time$t$
  • Velocity will be negative if the particle is moving in the negative direction
  • A velocity of zero means the particle is stationary$v=0$

• Speedspace open vertical bar v close vertical bar 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

    • ifspace v equals 4 comma space open vertical bar v close vertical bar equals 4
    • ifspace v=−6 comma open vertical bar v close vertical bar=6

• Acceleration$a$ m s^-2

  • The acceleration of a particle is the rate of change of its velocity at time$t$

  • 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 $a=0$ 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,$v=0$

  • a particle described as moving “due east” or “right” or would be moving in the positive horizontal direction

    • this also means that$v>0$

  • a particle “dropped from the top of a cliff” or “down” would be moving in the negative vertical direction

    • this also means that$v<0$

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

• Graph above the x-axis means the object is moving** forwards**

  • Graph below the x-axis means the object is moving backwards
  • The area between graph and the x-axis tells us the change in displacement **of the object

• 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

Exam Hint

• In an exam if you are given an expression for the velocity then sketching a velocity-time graph can help visualise the problem

Worked Example A particle is projected vertically upwards from ground level, taking 8 seconds to return to the ground.

The velocity-time graph below illustrates the motion of the particle for these 8 seconds.

i) How many seconds does the particle take to reach its maximum height? Give a reason for your answer.

ii) State, with a reason, whether the particle is accelerating or decelerating at time$t=3$.

*i) **At maximum height, velocity is zero. *

$v=0$ at $t=4$* *

Final answer: The particle takes 4 seconds to reach its maximum height. This is because its velocity is 0 ms-1 at 4 seconds.**

**

ii)

At $t=3$, velocity is positive

Acceleration is the gradient of velocity.

At $t=3$, acceleration is negative

Final answer: At 3 seconds the particle is decelerating as its velocity and acceleration have different signs.