Newton's Second law of motion: It states that rate of change of momentum is directly proportional to unbalance external force. And change of momentum is in the direction of unbalance force.

Mathematical Derivation: Initial momentum , Pi = mu

Final momentum , Pf = mv

Change in momentum ,∆p = Pf - Pi

= mv - mu

But we know that rate of change of momentum is directly proportional to the applied unbalanced force.

∴ F ∝ ∆p/t

or F ∝ (mv - mu ) / t/

Or F ∝ m. ( v -u )/t

Or F ∝ m. a , Where a= Acceleration

Or F = k. m. a ,

Or F = 1. m. a , In SI system K=1

∴ F = m. a

Mathematical Derivation: Initial momentum , Pi = mu

Final momentum , Pf = mv

Change in momentum ,∆p = Pf - Pi

= mv - mu

But we know that rate of change of momentum is directly proportional to the applied unbalanced force.

∴ F ∝ ∆p/t

or F ∝ (mv - mu ) / t/

Or F ∝ m. ( v -u )/t

Or F ∝ m. a , Where a= Acceleration

Or F = k. m. a ,

Or F = 1. m. a , In SI system K=1

∴ F = m. a