initial-state final-state diagram

A diagram illustrating the configuration of the system at the beginning and end of a specified period.

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.

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.

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:

Step 1. Identify the system and indicate it with a circle. (In this problem we will consider the system to be both cars. The reason is that we will assume that the collision time is very short so the possible external forces exerted on the system will produce a negligible external impulse, so we can solve the problem by using that the linear momentum of the two-car system is constant).

Step 2. Identify the initial state of the system. (In this problem, the initial state is right before the collision when the car of mass
m₁ is at rest and the car of mass m₂ is moving to the right with speed v₂(t_before). )

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).