state Newton's second law of motion. derive Newton's first law of motion from the mathematical formulation of second law of motion
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It states that the rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction in which the force acts. Mathematical formulation of Newton's Second Law of Motion: Let mass of an moving object be m. Let is initial velocity be u and final velocity be v
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Newton's Second law of motion :- The rate of change of momentum is directly proportional to the force applied on the system. Force applied is directly proportional to the product of mass and acceleration
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 The change in momentum∝ mv – mu The change in momentum∝ m × (v – u). The rate of change of momentum ∝ m × (v −u)/t Or, the applied force, F ∝m × (v −u)/t F = km (v - u)/t F = kma Here a [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. and F = ma which is the mathematical expression on the Newton's second law of motion.