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Motivation for Technique

While studying a given problem we are interested in what is happening to one or a set of objects that we call system. As time goes on we can observe changes in the system. For example, while pushing a box along a frictionless surface we observe that its speed increases. The velocity of the box has changed from some initial value at an initial time, v_i(t_i), to some final value at a final time, v_f(t_f). If we consider the box to be the system, the change in the system is measured by the change in the box's velocity. In other words, the box's velocity at a given time can describe the state of the system. We say that the system has changed from an initial state to a final state.

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The initial-state final-state diagrams are most useful in collision problems or in problems solved by using the Mechanical Energy and Non-Conservative Work model. In the following figure we present the initial-state final-state diagram of a two-car collision. Before the collision, the car of mass m₁ is at rest and the car of mass m₂ is moving to the right at speed v₂(t_before). After the collision, both cars are moving. The car of mass m₁ is moving to the right with speed v₁(t_after) and the car of mass m₂ is moving to the left with speed v₂(t_after). The goal of the problem is to find the speed of the car of mass m1 before the collision.

Constructing Initial-State Final-State Diagrams

To draw the diagram you must proceed as follow:

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Step 3. Identify the final state of the system. (In this problem, the final state is right after the collision when the car of mass m₁ is moving to the right at speed v₁(t_after), and the car of mass m₂ is moving to the left with speed v₂(t_after).