Uniform acceleration: When the change in the velocity of a moving body is the same in each second then its acceleration is said to be uniform. Such motion is said to be uniformly accelerated motion or uniform motion.
Non-Uniform acceleration: When the change in velocity of a moving body is not the same in each second then its acceleration is said to be non-uniform acceleration. Such motion is called non-uniform motion.
The equation of motions from the velocity-time graph:
The velocity-time graph for a body under uniform acceleration is shown in the figure.
Let initial velocity of the body = u
The final velocity of the body = v
Time is taken by the body = t
Acceleration of the body = a
Derivation of the first equation of motion:
According to the definition,
Acceleration of a body = Rate of change of velocity i.e. slope of the velocity-time graph.
The velocity-time graph for a uniformly accelerated body is given by the straight line AB. So, acceleration of the body is equal to the slope of the line AB.
Acceleration = Slope of line AB = \(\frac{BD}{AD}\) = \(\frac{BC-DC}{AD}\)
From the velocity time graph,
BC = v, DC = OA = u, AD = OC = t
Then, a ⇒ \(\frac{v-u}{t}\) ⇒ at
= v - u ⇒ v = u + at
⇒ First equation of motion
Derivation of the second equation of motion
Under the uniform acceleration, from the figure, one can write
Distance travelled (s) = Area of trapezium OABC
S = Area of the triangle ABD + Area of rectangle OADC
This is second equation of motion.
Derivation of third equation of motion
Distance travelled, s = Area of the trapezium OABC
S = \(\frac{1}{2}\)(Sum of the parallel sides) x Perpendicular distance between the two parallel sides
S = \(\frac{1}{2}\) x (OA + BC) x OC
S = \(\frac{1}{2}\)(u + v) x t But a = \(\frac{v-u}{t}\) ⇒ t = \(\frac{v -u}{a}\)
⇒ S = \(\frac{1}{2}\)(v + u) x \(\frac{v-u}{a}\)
⇒ S = \(\frac{v^2-u^2}{2a}\) ⇒ v2 - u2 = 2as
⇒ Third equation of motion
