Hello dear, in this article we are going to illustrate the important tool to study the parameters of motion like initial velocity, final velocity, acceleration and time with help of some equations known as equation of motion. These equations are valid for uniform motion only. The relation between initial velocity, final velocity and acceleration which are function of time can be given with the help of following equations which are,
1.) v = u +at
2.) s = ut + ½ at2
3.) 2as = v2 – u2
Where, u = initial velocity, v = final velocity
a = acceleration, t = time
Equations of motion can be derived in various ways, but here we are going to derive them using the graphical method.
Consider the body moving in XY plane starts its motion from point P with initial velocity ‘u’ (u≠0) and reaches to point Q in time ‘t’ with uniform acceleration ‘a’ and final velocity ‘v’ shown in figure below.
In graph X- axis represents time and Y-axis represents velocity,
From graph we have,
OP = u (initial velocity)
OQ’ = v (final velocity)
OS = t (time)
Draw QS perpendicular to X axis
Now draw PR ll OS
Then from graph, OS = PR, OP = SR and
OQ’ – OP = RQ = v – u = change in velocity
First equation of motion:
Acceleration is defined as the change in velocity per unit time, then from graph, slope of line PQ can also represents the acceleration,
Second equation of motion:
According to definition, velocity is rate of change of displacement hence,
∴ velocity = displacement/time
∴ displacement = velocity × time
From the above equation, we can say that the displacement (from point P to Q) of object is nothing the area covered object in the quadrilateral OPQS
∴displacement=area of quadrilateral OPQS
(□OPQS) is combination of rectangle and triangle so that
Third equation of motion:
Third equation of motion can be obtained by slight modification in mathematical steps used in derivation of second equation. Here also according to definition of velocity, distance covered by object is nothing but the area of quadrilateral OPQS. Here we are going to just use the formula for trapezium since quadrilateral OPQS acts as trapezium if the entire diagram is rotated by 900 in clockwise sense.
We know that,