# MATHEMATICAL FORMULATION OF SECOND LAW OF MOTION

MATHEMATICAL FORMULATION OF SECOND LAW OF MOTION

Suppose an object of mass, m is moving along a straight line with an initial velocity, u. It is uniformly accelerated to velocity, v in time, t by the application of a constant force, F throughout the time, t. The initial and final momentum of the object will be, p1 = mu and p2 = mv respectively

The change in momentum ∝ p2 – p1 ∝ mv – mu
∝ m × (v – u).
The rate of change of momentum ∝ m (v -u) / t
Or, the applied force, F ∝ m (v – u ) / t
F= k m a eq 1
Here a = [ (v – u) / t ] is the acceleration,

which is the rate of change of velocity. The quantity, k is a constant of proportionality.
The SI units of mass and acceleration are kg and m s-2 respectively. The unit of force is so chosen that the value of the constant, k becomes one. For this, one unit of force is
defined as the amount that produces an acceleration of 1 m s-2 in an object of 1 kg mass.

That is,

1 unit of force = k × (1 kg) × (1 m s-2).
Thus, the value of k becomes 1. From Eq. 1
F = ma
The unit of force is kg m s-2 or newton, which has the symbol N. The second law of motion gives us a method to measure the
force acting on an object as a product of its mass and acceleration.