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Two identical perfectly elastic spheres slide towards one another on a smooth horizontal surface. The velocities of the spheres are vx and v2. What are the velocities of the spheres after a direct collision? The collision is perfectly elastic. There is no friction. 

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Answer: The spheres exchange velocities after the collision.

Explanation:

If the masses of the spheres are m1 and m2, and we denote their velocities after the collision by x and y, then, according to the law of conservation of momentum,

m1v1 + m2v2 = m1x + m2y. (1)

 Using the law of conservation of energy and assuming that the total kinetic energy of the spheres is the same after the collision, we may write

1/2(m1v12) + 1/2(m2v22) = 1/2(m1x2) + 1/2(m2y2)   (2)

Solving equations (1) and (2) and setting m1 = m2 = m, we obtain y = v1 and x = v2 , i.e., in a perfectly elastic collision, identical spheres behave as if they exchange their velocities. If, before the collision, the first sphere was moving from left to right with the velocity v1? then after the collision it will move in the opposite direction with the velocity v2

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