| About Lessons Notes Practice Reference Contact Tropic of Calc |
Displacement, Velocity, and AccelerationPosition is a quantity of which we cannot give a good definition. We can describe it in terms of coordinate systems and transformations, but it is very hard to analytically define without using recursion. Time is much the same. We begin our study of mechanics—the study of forces and motion—with the two concepts of position and time. We must rely on our intuitive understandings of those concepts. We denote position with a variety of letters: Displacement is defined as the vector change in position and denoted by putting a Perhaps an example will make things clearer. MapQuest gives the distance from Old Orchard to New Trier as approximately 6.39 km. Therefore, if we make the round trip from Old Orchard to New Trier and back, we have traveled a distance of Velocity is the displacement per unit time. We notate velocity with Acceleration is the change in velocity per unit time and is notated Each of these derivatives of position can be treated as scalar quantities as well. Only velocity has a separate name for the scalar analogue: speed, defined as In our next lesson, we will discuss motion with constant acceleration in one and several dimensions.
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, 
, 
, 
, 
, and 
are the most common ones. We will typically use 
, the Greek letter Delta, in front of the position variable. For example, 
where 
is the final position and 
the initial position. Distance is a different concept: it is the scalar change in position. It is also denoted by putting a 
. Our displacement 
, however, is 0. km because we are in the same place as we started; a vector drawn from our initial position to our final position has a length of zero.
, so 
. If we want an instantaneous velocity, we take the velocity over ever-smaller time intervals 
. In other words, 
. This looks suspiciously like a derivative; in fact, it is. Therefore we can equivalently state 
.
, so 
. Instantaneous acceleration is calculated similarly as 
.
.