Question

derive second equation of motion.

Answer 1

 Second equation of motion:
s = ut + 1/2 at^2

sol.
Let the distance travelled by the body be “s”.
We know that
Distance = Average velocity X Time

Also, Average velocity = (u+v)/2

.: Distance (t) = (u+v)/2 X t …….eq.(1)

Again we know that:
v = u + at
substituting this value of “v” in eq.(2), we get

s = (u+u+at)/2 x t
=>s = (2u+at)/2 X t
=>s = (2ut+at^2)/2
=>s = 2ut/2 + at^2/2

or s = ut +1/2 at^2

This is the 2nd equation of motion.

Answer 2

we know 

v = ds/dt 

ds = vdt = (u + at)dt 

when time = 0 , distance travelled = 0 

when time = t , distance travelled = s (say )

Integration , 

∫ds = ∫(u + at) dt = u∫dt + a∫t dt 

s = 0 = u(t-0) + a [t²/2 - 0 ] 

s = ut + 1/2 at²

This is second equation of motion