Two practically elastic spheres collide. If their velocities before impact have same magnitude and opposite directions and one has a mass 3 times the other, what are their velocities after impact?

The collision of two spheres can be analyzed using the principles of conservation of momentum and kinetic energy.

1. Conservation of Momentum: The total momentum before the collision is equal to the total momentum after the collision. If we denote the mass of the smaller sphere as m and its velocity as v (before the collision), and the mass of the larger sphere as 3m (since it's three times the mass of the smaller one) and its velocity as -v (since they are moving in opposite directions), then the total momentum before the collision is mv - 3mv = -2mv. After the collision, if we denote the velocity of the smaller sphere as v1 and the velocity of the larger sphere as v2, then the total momentum after the collision is mv1 + 3mv2. Setting these equal gives us the equation mv1 + 3mv2 = -2mv.

2. Conservation of Kinetic Energy: The total kinetic energy before the collision is equal to the total kinetic energy after the collision. The kinetic energy before the collision is 1/2 * m * v^2 + 1/2 * 3m * (-v)^2 = 2mv^2. After the collision, the kinetic energy is 1/2 * m * v1^2 + 1/2 * 3m * v2^2. Setting these equal gives us the equation v1^2 + 3v2^2 = 2v^2.

Solving these two equations simultaneously gives v1 = -v and v2 = v. This means that after the collision, the smaller sphere is moving in the opposite direction with the same speed, and the larger sphere is moving in the original direction of the smaller sphere with the same speed.