Kinematics Toolkit

Displacement, Velocity & Acceleration

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 $v=0$

• Speed $\left|v\right|$ 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$v=4, \left|v\right|=4$

    • if$v=-6, \left|v\right|=6$

• 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,$\mathbf{ }\mathit{v}\mathbf{=}\mathbf{0}$

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

    • this also means that$\mathbf{ }\mathit{v}\mathbf{>}\mathbf{0}$

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

    • this also means that$\mathbf{ }\mathit{v}\mathbf{<}\mathbf{0}$

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