B.5.QD:5.2 - From a position to a sufficient state or useful bound

A model describes a point moving along one line in an inertial frame. During the next two seconds there is no net force; use the classical relation q(t) = q(0) + v(0)t. The supplied position is q(0) = 0. Someone asks where the point will be two seconds later.

Construct two cases allowed by this description. With v(0) = +1 metre per second, q(2) = +2 metres. With v(0) = -1 metre per second, q(2) = -2 metres. Position alone leaves the requested answer undetermined.

One next question is what information completes the state for this prediction? Under this model, initial velocity together with position suffices. Obtaining velocity could use an already available displacement over a known interval of constant velocity. A.3.3 and C.16 develop state and measurement when those contributions are needed.

Another question is cheaper if the current decision only asks whether the point stays within three metres of its starting position for the next two seconds. Suppose the available information bounds the initial velocity between -1 and +1 metre per second. Then the displacement magnitude is at most (1 metre per second)t throughout that interval, so the point stays within two metres. The bound answers the decision without acquiring a more specific velocity.

The two questions support different uses. If an actuator must meet the point at a specified position, the bound can be insufficient and the state question becomes useful. If the net force becomes nonzero during the interval, revise the model and its prediction. The construction of the question has located the relevant missing contribution in each case.