Force and Acceleration
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"`
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/s2 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/s2 in an object of 1 kg mass.
1 unit of force = k × (1 kg) × (1 m/s2).
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.
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