Equations of motion

The three equations of motion give a relation between various physical quantities dealing with state of motion of an object.
If we consider a body having some initial velocity u.It attains a velocity v in time t by having a continuous acceleration a.
The three  equations of motion are:

Velocity - Time Relation
v=u + at
Position -Time Relation
s = ut + (1/2) at^2
Position - Velocity Relation
v^2  - u^2 = 2 as

Where s denotes the position of the object.

They can be derived with the help of a v-t graph

Motion

I will be mainly dealing with the most basic way to make you understand the concept and will not elaborate on the same thing .... we have textbooks and help books for those .This mainly covers CBSE syllabus for class 9.So here we go.....

What is motion ?

An object is said to be in motion if it changes position with time with respect to a stationery object.
The frame of reference is very important .

Nothing in this universe can actually be said to at rest since we know the universe is continuously in a state of expansion and everything in it is along with it is moving .

But ....we can say the person sitting next to us is at rest when our frame of reference is the building and trees nearby.

To understand motion we need to understand some terms like  distance , displacement ,speed ,velocity ,acceleration etc. which can be understood more clearly if we know what are vector and scalar quantities.Any measurable quantity can be termed as a physical quantity.

Vector Quantities:

Physical quantities which are fully described only if the magnitude as well as the direction of change needs to be specified.For example if you are telling someone location  of your home you specify directions along with the length of path to be covered .Examples will be covered later once we have introduced the terms.

Scalar quantities:

Physical quantities which are fully described by their magnitude alone .For example if you ask someone to get chocolates , you simply specify how many chocolate you want .No North , south .....associations needed.


Now that we understand scalar and vector quantities we can better understand distance and displacement .

Distance :

Distance between any two points is the length of the path covered between initial and final position of the object .It is a scalar quantity. Its SI unit is m

Displacement :

The shortest straight line path between initial and final position of an object in a specified direction.It is a vector quantity.Its SI unit is m .We need to specify direction.

Speed :

It is the rate of change of distance .It is a scalar quantity.

Speed= Distance / Time

SI unit : m/s

Velocity: 

Its the rate of change of displacement.Its a vector quantity.

Velocity = Displacement/ Time

SI unit: m/s

Uniform motion : 

If the speed of an object remains same throughout its state of motion the object is said to be in uniform motion i.e the object covers equal distance in equal intervals of time and vice versa.In data form it would look like this: 

Distance (in m)        Time (in s)

10                                2

20                                4

30                                6

40                                8

50                                10

  If we plot a distance -time graph for this  data ,it would be a straight line  passing through the origin with time on x axis and distance on y.

Distance Time Graphs

• The change in the position of an object with time can be represented on the distance-time graph.We can get to know so many things if we know how to read a graph.
• In this graph, time is taken along the x-axis and distance is taken along the y-axis.
• Distance time graphs of a moving body can be used to find the speed of the body as they specifically represent velocity.
• The distance time graph for a body moving at uniform speed is always a straight line as distance travelled by the body is directly proportional to time as shown above.
• The distance time graph for a body moving with non uniform velocity is a curve as shown below:

• The distance time graph is parallel to time axis when the object is at rest .
  To calculate speed of the body from distance time graph say at point A first draw a perpendicular AB on time axis and a perpendicular AC on distance axis.
• on distance axis so that AB represents the distance travelled by the body in time interval OB and since we know thatSpeed = Distance / Time =AC/ AB

• Velocity time graphs

• The variation in velocity with time for an object moving in a straight line can be represented by a velocity-time graph.
• In this graph, time is represented along the x-axis and the velocity is represented along the y-axis.
• The product of velocity and time give displacement of an object moving with uniform velocity. The area enclosed by velocity-time graph and the time axis will be equal to the magnitude of the displacement.
• If a body moves with a constant velocity then velocity time graph for this body would be straight line parallel to time axis 
  The velocity time graph of uniformly changing velocity  is a straight line. We can find out the value of acceleration using the velocity time graph.
• For calculating acceleration at time corresponding to any point 
• a= Change in velocity given by axis y/Time taken given by axis x 
  
  which is equal to the slope of velocity time graph. So we conclude that slope of velocity time graph of moving body gives its acceleration.
• The distance travelled by moving body in a given time will be equal to area of triangle as 
  Area of triangle = 1/2 × Base × Height
• When the velocity of a body changes in an irregular manner then velocity time graph of the body is a curved line.

• Velocity time graph of an object moving with uniform decreasing velocity:

  The slope of the velocity time graph of an object moving with uniform decreasing velocity with uniform acceleration is a downwards straight line. The straight downward slope shows the decreasing velocity with uniform acceleration, i.e. retardation.
  Velocity time graph of an object moving with non-uniform velocity:
  Zig–zag line of slope of graph shows that the object is moving with non-uniform velocity.