Position, velocity, & acceleration

Position (`vec d`) is a vector quantity representing a location relative to an origin, usually measured in metres (m). A change in position is called displacement (`Delta vec d`). The rate of change in position per unit of time is called velocity (`vec v`), usually measured in metres per second (m/s).

If we graph position versus time (or d–t for short), the slope of the tangent line gives us the instantaneous velocity at that moment. The slope of the secant line gives us the average velocity on the interval [`t_1`, `t_2`]. We can also go the other way: on a v–t graph, the area under the curve between `t_1` and `t_2` gives us the displacement on that interval.

tv(t1,v1)(t2,v2)dtd(t1,d1)(t2,d2)vav
Graphs of position vs. time and velocity vs. time

Vectors can be represented in two ways. In handwriting, we always use an arrow, like `vec v`. In other situations, boldface can be used as an alternative to the arrow, like v.

Acceleration (`vec a`) is the rate of change of velocity per unit of time, usually measured in metre per seconds squared (m/s2). It relates to velocity in exactly the same way that velocity relates to position. In this course, we always assume that acceleration is constant.

tv(t1,v1)(t2,v2)aavta(t1,a1)(t2,a2)v
Graphs of velocity vs. time and acceleration vs. time

There are two important equations that are represented in these two sets of graphs. When we are calculating slopes, we use

`vec v_"av" = (Delta vec d)/(Delta t) qquad and qquad vec a_"av" = (Delta vec v)/(Delta t)`,

and when we are going the other way by calculating areas, we use the equivalent equations

`Delta vec d = vec v_"av"Delta t qquad and qquad Delta vec v = vec a_"av"Delta t`.

We need to identify a couple identities to help us use these two equations. For any variable `x`, we assume that

`Delta x = x_2 - x_1 qquad and qquad x_"av" = (x_1 + x_2)/2`.

We now have all we need to solve any motion problem. However, some calculations are so common that it is worth committing them to memory. In the next section, we’ll derive a few other, time-saving equations.