A Level Maths: Mechanics - An Introduction To Kinematics
Introduction
As an introduction to kinematics, we will be looking at how objects generally move in a straight line and we will ignore the size and shape of these objects.
In order to understand this topic fully there are some mathematical definitions that need to be introduced.
Distance – this is how far you travel
Speed – this is how fast you go
Displacement – this is the position of an object relative to a fixed point. For instance if you consider a starting point and walk 5m away and then walk back the same 5m. Then the distance travelled will be a total of 10m but the displacement will be 0m because you are back where you stated.
Velocity – this is defined using the speed of an object and its direction
Average speed – often referred to as the constant speed that may have been travelled to complete a journey
How Kinematics Works in A Level Maths
Example:
The position of a particle is given by x=2+t(t-3)
a) Find the position of the particle at times t=0, 1, 1.5, 2, 3, 4,5
b) Find the displacement of the particle relative to initial position when t=5
c) What is the total distance travelled?
Solution
a) Here we have a table of values showing the position at the times requested:
b) From the table of values it can be seen that at t=5, x=12 and at t=0, x=2. The displacement is found by 12 – 2 = 10m
c) The total distance travelled can be found by find the difference in x between each time i.e. 2, 0.25, 0.25, 2, 4, 6 and by adding these we obtain 14.5m
Introduction to Kinematics - Speed and Velocity
Speed is a scalar quantity and does not involve direction. Velocity is a vector because it has size and direction.
\begin{aligned} \text { Average speed } & =\frac{\text { Distance Travelled }}{\text { Time Taken }} \\ \text { Average Velocity } & =\frac{\text { Change in displacement }}{\text { Time Taken }} \end{aligned}Displacement Time Graphs
It is possible to plot a graph showing displacement against time. These are known as displacement time graphs. The gradient of such a graph gives the velocity and the graph can be a straight line or a curve.
Velocity Time Graphs
As the name suggests, this will show a graph of velocity against time. Quite often velocity time graphs show a particle accelerating, maintaining a constant speed before decelerating before coming to rest.
Acceleration
Acceleration is often used to describe the motion of cars, bikes and planes. It is the rate at which the velocity is changing and the units are ‘metres per second per second’ or m s^{-2}
An acceleration of 5 \mathrm{~ms}^{-2} means that the velocity is changing 5 \mathrm{~ms}^{-1} every second.
\text { Average acceleration }=\frac{\text { Change in velocity }}{\text { Time }}
Introduction to Kinematics – Using areas to find distances and displacements
The area between a velocity time graph and the horizontal axis represents the distance travelled.
Most graphs will consist of straight lines and it is generally best to divide the area into shapes that are easy to identify such as rectangles, triangle and rectangles.
Example
A runner jogs from A to B starting at rest and then accelerates uniformly to 8m s^{-1}
in 8 seconds. He maintains this speed for 20 seconds and then comes to rest at B. The whole journey takes 40 seconds.
a) Sketch a velocity time graph
b) What is the acceleration in the first 8 seconds?
c) What is the acceleration in the final part of the journey?
d) What is the total distance ran?
Solution
a) The velocity time graph is shown below
b) \text { Acceleration }=\frac{8}{8}=1 \mathrm{~ms}^{-2}
c) \text { Acceleration }=-\frac{8}{12}=-\frac{2}{3} m s^{-2}
d) \text { Distance travelled }=\text { Area under curve }=\frac{1}{2}(8)(20+40)=240 m
Even though this is an introduction to kinematics the applications are quite vast and another important area is that of SUVAT which will be explored in another blog post.