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Force and Acceleration

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**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" xx "(v - u)"/"t"`

or, the applied force,

`"F" ∝ "m"xx"(v - u)" / "t"`

`"F" ∝ "km"xx"(v - u)" / "t"`

=kma

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.

**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.