# Kinematic variables

#### Prerequisites

The key idea in describing motion is where and when. If we set up a coordinate system to be able to describe position and can specify position with a vector, then all we have to do to describe motion is tell where something is (its position) as a function of when we are looking at it (the time). We write this in 2D with a vector where both the vector and the specifying position coordinates depend on (are functions of) the time. We write it this way:

$$\overrightarrow{r} = x(t)\hat{i} + y(t)\hat{j}$$

This gives us the *value* of the position. We now have to explore how it changes and at what rate it changes.

The position, change in the position, and the various rates of change of position (its derivatives) are referred to as *kinematic variables*, since "kinematics" means "the study of motion", from the Greek word for movement, *kinesis*.

## Displacement

The basic concept is the change of position. This is referred to as a *displacement*. As usual, we will write a change in a quantity with the change marker Δ. Thus, the displacement an object undergoes from time *t*_{1} to time *t*_{2} is given by $\Delta\overrightarrow{r} = \overrightarrow{r}(t_2) - \overrightarrow{r}(t_1)$ .

If this is written out in terms of coordinates and regrouped, we can see how this is represented in terms of x and y coordinates:

$$\Delta\overrightarrow{r} = \overrightarrow{r}(t_2) - \overrightarrow{r}(t_1)$$

$$\Delta(x\hat{i} + y\hat{j}) = (x(t_2)\hat{i} + y(t_2)\hat{j}) - (x(t_1)\hat{i} + y(t_1))\hat{j}$$

$$(\Delta x)\hat{i} + (\Delta y)\hat{j} = (x(t_2) - x(t_1))\hat{i} - (y(t_2) -y(t_1))\hat{j}$$

Since the $\hat{i}$ and the j$\hat{j}$ point in different directions, if we have an equation with both i's and j's in it, the coefficients of the i's have to be equal and the coefficients of the j's have to be equal. A displacement in the x direction can never cancel a displacement in the y direction. So we can conclude that the things multiplying the i must be equal and the things multiplying the j must be equal:

$$\Delta x = x(t_2) - x(t_1)$$

$$\Delta y = y(t_2) - y(t_1)$$

This kind of algebraic re-grouping is extremely valuable in physics. The reason we rearrange is that in the different arrangements we are focusing on different ways of identifying what the physical meaning is. When we write it in terms of the entire vector, we are thinking about the shift from the initial position to the final. When we break it up in terms of coordinates, we are focusing on how each coordinate changes. Of course the result of the total displacement will be the same, but the coordinate analysis gives us an additional understanding of the structure.

Note that the position vector can be thought of as the displacement with respect to the reference point (origin) of the coordinate system. But since we can't specify a position without an origin or reference point, we tend to talk about "position" and "displacement" differently.

## Velocity

Now we can look at how fast the change occurs. Dividing a displacement by the time it takes to make that displacement tells us not just a change, but a *rate of change*. That's much more useful. How fast you're traveling from here to there can make the difference whether you come late for the concert or maybe miss it entirely. The rate at which a chemical reaction takes place can make the difference whether a cell lives or dies.

The rate of change of position is called the *velocity*. There are some tricky aspects of thinking about velocity, so read the follow-on page giving the details.

## Acceleration

Once we have a changing position we can get the velocity. But the velocity may now change, so we can look at the rate of change of the velocity. This is called the *acceleration*. This turns out to be of immense importance in physics because of the principle that relates motion to the factors that result in changes in an object's motion -- the forces. Acceleration also is tricky to think about. While we naturally see velocity, seeing acceleration in a motion takes some attention -- and some analysis. Read the follow-on page giving the details.

## Jerk and higher rates of change

Of course if the position is changing and the velocity is changing, the acceleration could be changing as well. The rate of change of acceleration is called the *jerk*. (Really!) We could go to even higher derivatives, but it's been found to be rarely necessary in describing motion. An interesting example is in thinking about the spinning earth. This involves the analysis of circular motion, so read about that before trying the problem, "I felt the earth move".

#### Follow-ons

Last Modified: July 12, 2019