Equations of motion by graphical method

1. First equation of motion:- The first equation of motion is : v = u + at. It gives the velocity acquired by a body in time t.

Acceleration = Change in velocity/ Time taken

or, Acceleration = Final velocity – Initial velocity/ Time taken

a = v – u/ t

at = v – u

v = u + at

where, v = final velocity of the body

u = initial velocity of the body

a = acceleration

t = time

2. Second equation of motion:- The second equation of motion is s = ut + 1/2at² . It gives the distance travelled by a body in time t.

Average velocity = Initial velocity + Final velocity/ 2

that is, Average velocity = u + v/2

Distance travelled = Average velocity × Time

s = ( u + v )/2× t

From the above equation of motion we have v = u + at. Putting the value of v in above

s = ( u + u + at)/2 X t

s = ( 2u + at)/2 × t

s = 2ut + at²/ 2

s = ut + 1/2at²

where, s = distance travelled by the body in time t

u = Initial velocity of the body

a = acceleration

3. Third equation of motion:- The third equation of motion : v²= u² + 2as. It gives the velocity acquired by a body in travelling a distance s.

From the second equation of motion we have: s = ut = 1/2at²…………………(1)

And from the first equation of motion we have: v = u + at ………………………..(2)

so, at = v -u

or t = v – u/a

putting the value of t in equation ( 1) we get:

s = u(v-u)/a + 1/2× a ( v-u/a )²

s = uv – u²/a + a( v² + u² – 2uv)/2a² (because =(v-u)² = v² + u² – 2as)

s = uv – u²/a + v² + u² – 2uv/2a

s = 2uv – u² + v² + u² – 2uv/ 2a

or 2as = v²– u²

v² = u² + 2as

where, v = final velocity

u = initial velocity

a = acceleration

s = distance travelled