If we taught motion like we often teach energy, it might look something like this.
We say that displacement is done when a velocity persists through an amount of time. Quantitatively we can define displacement as D = V Δt. This equation is valid as along as the velocity persisting through that time is constant. For non-constant velocity persisting through time, see the appendix.
Given this definition of displacement, we can now proceed to talk about the change-in-position-displacement theorem, which states that when a displacement occurs, the amount of displacement is just equal to the change in position. For a proof of this theorem, see the appendix. The change-in-position-displacement theorem can be written as the following equation,
D = Δx
where Δx is understood to be equal to xf -xi. It is a surprising result that the act of displacing an object with a velocity through a time is merely equal to the change in position.
Together the definition of displacement above and the change-in-position-displacement theorem allow us to solve many problems involving moving objects. Before we do so, however, we need to discuss a special kind of velocity called a conservative velocity. A conservative velocity is one that can be defined as the negative time-derivative of a displacement function. When such a function exists, mechanical displacements are said to be conserved. For this reason, we say that space can never be created or destroyed.
The conservation of space makes it very convenient to solve problems about motion, because we can now define a number line in space with an arbitrary zero-point. It may be confusing to think about negative position, but remember only changes in position through space are important.
Problem #1: A velocity of 4 m/s persists backwards in time for 4s. What is the displacement? (Hint: The velocity is persisting in the opposite of time’s conventional flow, so be careful with your signs!)
Problem #2: Apply the principle of conservation of space to determine how far an object has displaced if it started at the arbitrarily defined 5 m position, and underwent a displacement of 6 m.
Problem#3: When an object moves at with a velocity of -12.8 m/s for 4 s, how much of the displacement is stored in potential space?