prove the formula for the preservation of linear momentum from Newton's third law of motion.
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Answer:
Newton's third law states that every action has an equal and opposite reaction. Its prove is given above. It is done through Newton's 2nd law of motion. Principle of conservation of linear momentum states that,
If no external forces act on the system of two colliding objects, then the vector sum of the linear momentum of each body remains constant and is not affected by their mutual interaction.
Let us consider an isolated system of n particles having initial momentum p1, p2 ... pn. Due to the collision, let the momentum of the particles after collision be p1', p2' ... pn' respectively. Then according to the principle of conservation of linear momentum, in the absence of external force,
p1 + p2 + ... + pn = p1' + p2' + ... + pn'
Momentum of a body is product of mass and velocity
Fig: Collision between two spheres moving in same direction in a straight line.
To prove this principle, we consider a collision between two spheres A and B having masses of m1 and m2 respectively. Let u1 and u2 be the velocities of the spheres before collision such that u1 > u2 and moving on the same straight line as shown in the figure. After collision, let their velocities be v1 and v2 on the same line. If they collide with each other for a short time interval t, each sphere exerts a force on the other sphere and so, the force experienced by A is given as
Similarly, force experienced by B is
According to Newton's third law of motion, the force experienced by A and B are equal and opposite,
or,
That is, total momentum before collision is equal to total momentum after collision if no external forces act on them which proves the principle of conservation of linear momentum.
Explanation:
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