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, vi(ti), to some final value at a final time, vf(tf). 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.
It is always useful to sketch the initial and final state of the system. The following figure is what we call an initial-state and final-state diagram. The dotted circle indicate the system under consideration.
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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 cars 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(tbefore). After the collision, both cars are moving. The car of mass m1 is moving to the right with speed v1(tafter)_m and the car of mass _m2 is moving to the left with speed v2(tafter).The goal of the problem is to find the speed of the car of mass m1 before the collision.
To draw the diagram you must proceed as follow: