Question

using second law of motion derive the relation between force and acceleration

Answer 1

According to second law of motion Force is directly proportional to rate of change of momentum.
so F is directly proportional to dp/dt
we know that momentum P= MV
so
F= d(mv)/dt=m(dv/dt)
thus we get F=ma.

Answer 2

According to second law of motion
F=change in momentum per unit time
F=∆p/∆t…………………(1)

As we know p=mv
So change in momentum (∆p)=∆(mv)
∆p=m∆v.................... (2)
(since mass is constant and only velocity changes)

Also acceleration a=change in velocity per unit time
a=∆v/∆t…………………(3)

Using (1)&(2)
F=m∆v/∆t
Using (3)
F=ma

This is the relation between force and acceleration which you are looking for