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Illustration 8.3: Hard and Soft Collisions and the Third Law

m1 = kg, m2 = kg

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This Illustration models collisions between two particles (position is given in meters and time is given in seconds). Restart. Both animations assume identical particles and both animations start with the particles having the same velocities. What is different about the animations is the interaction between the two particles. The interaction in Animation 1 can be characterized as hard since the acceleration is very large and the interaction is very short range. It is also called a contact interaction because the force turns on when the two particles are in contact.

In Animation 2 the interaction can be characterized as soft. Vary the masses of the two particles and be sure to pay attention to the scale of the acceleration graph in each animation. In addition, watch the relative acceleration of the particles as you vary the masses. Notice that the accelerations are different when the masses are different.

What can you say about the force experienced by each particle in each animation? The character of the forces is different: One is soft and one is hard. Nonetheless, the forces are always equal and opposite. To see this you must take each object's acceleration and multiply it by its mass. This is exactly the statement of Newton's third law, the law of reciprocity of forces. In this case, there are no net external forces acting on the two particles, so the change in momentum of the two-particle system is zero. In other words, momentum is conserved. Using equations, we would say that since Σ Fnet = Δp/Δt or Σ Fnet = dp/dt, if the net force on a system is zero, then Δp/Δt = 0 or dp/dt = 0, which means that the change in momentum over time must be zero. Hence the sum of the two impulses experienced by the balls must be zero. If one particle's momentum goes up, the other particle's momentum must go down by exactly the same amount. Check it out by looking at the tables.

Two-dimensional models show a dramatic difference between hard and soft collisions. (See Problem 8.12 for two-dimensional collisions.) Hard collisions tend not to have much of an effect on incident particles except for the occasional particle that suffers a head-on impact. Soft collisions, on the other hand, produce minor deflections on a large number of particles. The experimental observation of alpha particles being deflected backwards from gold foil led Ernest Rutherford to predict that atoms have a small hard core, the nucleus.

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