**Theory:**In an elastic collision, both momentum (*p*) and kinetic energy (*T*) are conserved. That is **p**_{f} = p_{i} and **T**_{f} = T_{i}
Newton's pendulum is five metal spheres suspended so they all touch one another in line. One or more spheres can be drawn back and released. The released spheres swing down and collide with the remaining spheres. Each sphere has mass *m*, and the spheres collide essentially elastically. Of course, the fact that the spheres click when they connect means that some kinetic energy is being converted into sound energy and lost, but these losses are negligible in a single collision. When one or more spheres are drawn back and released, it/they have mass *M*_{i} = m, 2m, 3m, 4m,depending on if one, two, three, or four spheres are drawn back. At the instant just before these spheres collide with the other hanging spheres, they have a velocity (in the horizontal direction) of *v*_{i}. The momentum and kinetic energy are thus *p*_{i} = M_{i}v_{i} , and **T**_{i} =**(1/2)M**_{i}v_{i}^{2}. An instant after the spheres collide, some mass of spheres (*M*_{f}) must keep moving with velocity (in the horizontal direction) *v*_{f}. Applying the conservation of momentum to this collision gives the following equation: *p*_{f} = p_{i}
*M*_{f}v_{f} = M_{i}v_{i}
*M*_{f} = M_{i}(v_{i}/v_{f})
Conserving kinetic energy gives.... *T*_{f} = T_{i}
*(1/2)M*_{f}v_{f}^{2} = (1/2)M_{i}v_{i}^{2}
*M*_{f} = M_{i}(v_{i}/v_{f})^{2}
The red and green equations above can be combined to eliminate *M*_{f}. **M**_{i}(v_{i}/v_{f})^{2} = M_{i}(v_{i}/v_{f})
*v*_{i} = v_{f}
So the velocity of any moving spheres after the collision is the same as the velocity of the moving spheres before the collision. Substituting this into the red equation gives... *M*_{f} = M_{i}(v_{i}/v_{f})
*M*_{f} = M_{i} (v_{i}/v_{i})
*M*_{f} = M_{i}
The number of spheres moving after the collision is the same as the number moving before the collsion. |