Reading the content in the kinematic equations


In our description of the motion of an object the key idea was "Where and when?" — What was the position of the object and at what time was it there? How the object changes position is crucial for understanding the causes of motion. For this purpose we introduce average and instantaneous changes as coded in equations with deltas (Δ) and derivatives. Since our brains are not well designed to pick up on the details of how things move (though it's pretty good at figuring out where something is going), learning to code these details into equations — and pry them out again — is extremely useful.

We refer to equations like the kinetic equations below, that help you organize a whole subject conceptually, as anchor equations. They're immensely valuable, once you learn to see the physics in the equations. In these "Reading the content in ..." pages, we'll try to help you learn to do this.

The kinematic equations for motion in a single dimension (along a straight line) are:

[A] $\langle v \rangle = \frac{\Delta x}{\Delta t}$             [B] $ v = \frac{dx}{dt}$

[C] $\langle a \rangle = \frac{\Delta v}{\Delta t}$             [D] $ a = \frac{dv}{dt}$

The first row (A and B) tell us how the position changes (velocity); the second row (C and D) tells us how the velocity changes.

The first column (A and C) tells us how things change over finite (non-zero) time intervals; the second column (B and D) tells us how things change at an instant (at a specific time -- essentially zero interval).

The way information is coded in these equations can be a bit tricky as there are some standard conventions that often are not made explicit. These conventions can make them look a bit different from the math equations they correspond to. Here are some of the ways the symbols in the equations convey information.

1. v and a -- The symbols $v$ and $a$ represent the velocity and acceleration of the object we are describing. (And the symbols are chosen to represent the first letters of each of the quantities.) Although this is fairly explicit, what is hidden is that both of these quantities are considered to be not values but functions of time. If we were in a math class, we would probably write them as $v(t)$ and $a(t)$rather than just as $v$ and $a$.

2. <  > -- The brackets around the $v$ and $a$ in equations A and C indicate that we are considering the average of the quantity contained inside them. An average implies an average over some interval.  Therefore, the equations containing this symbol implicitly refers to a time interval, not just a single time. 

3. Δ -- Equations A and C also include the symbol Δ (capital delta = Greek "D" for "difference"). This symbol, like the brackets above, do not represent a quantity but say that you are going to do something to a quantity. While the brackets surround the quantity they modify, the delta modifies the next symbol. Specifically, it means that you are to consider the change in the symbol that follows it, not the value of that symbol. Since the brackets already tell us that we are referring to a time interval, the delta implicitly refers to that same interval. Thus, if the time interval we are considering starts at $t_1$ and ends at $t_2$ (often written $t_i$ and $t_f$ to represent initial and final times), the average would be taken over the time interval $t_1 → t_2$ and the combination $\Delta x$ would mean "change in $x$" and would represent $x_2 - x_1$. This makes sense since if you change something, you add something to it (or subtract if the quantity is negative): so changing $x_1$ by an amount $Δx$ means making a new value $x_2 = x_1+ Δx$ and this rearranges to give $Δx = x_2 - x_1$.

Making the two times more explicit makes the equations A and C look like this:

[A] $\langle v \rangle_{t_1 → t_2} = \frac{x_2 -x_1}{t_2-t_1}$            

[C] $\langle a \rangle_{t_1 → t_2} = \frac{v_2 -v_1}{t_2-t_1}$             

In this form, the dependence of the expression on the position and velocity at two times is explicit.

4. d/dt -- This cluster of symbols is used in a physics class to represent the derivative or rate of change. Explicitly, we read an expression like "$dx/dt$" as "the derivative of the function $x(t)$ with respect to $t$". (As remarked in 1 above, the dependence of $x$, $v$, and $a$on $t$ is often not explicitly indicated.) This particular notation is Leibniz's notation for the derivative. Newton put a dot above the function to indicate a derivative. In math classes you may see a prime (f' ) or a D (Df) to represent a derivative. The derivative of a function is also a function, so equations B and D are evaluated at a single instant of time. You can see a detailed discussion of this in the pages Instantaneous velocity and Example: Velocity at the top.

Leibniz's form is favored in physics classes for two reasons. First, it makes the connection with the Delta notation: the derivative is like an average velocity over a very small time interval. Second, since it looks like a ratio, it gives you correct guidance as to what the units of the derivative will be. Thus, $dx/dt$ has units of distance/time just like $\Delta x/\Delta t$ does.

5. → : vectors -- When we consider motion in more dimensions than just along a straight line our equations look like this:

[A] $\langle \overrightarrow{v} \rangle = \frac{\Delta \overrightarrow{r}}{\Delta t}$             [B] $ \overrightarrow{v} = \frac{d \overrightarrow{r}}{dt}$

[C] $\langle \overrightarrow{a} \rangle = \frac{\Delta \overrightarrow{v}}{\Delta t}$             [D] $ \overrightarrow{a} = \frac{d \overrightarrow{v}}{dt}$

This means that instead of four equations for the set of variables ($x, v, a$), we have for equation A, B, C, and D for each of the triples: ($x, v_x, a_x$), ($y, v_y, a_y$) and ($z, v_z, a_z$). There is a set for the motion in each dimension and they are independent of each other.

When do these equations hold? These equations are definitions so they always hold. (This is why we like to always use these forms rather than the forms often shown in physics texts as "kinematic equations". Those only hold for the case of constant acceleration. Our exact equations can give those very easily if you know how the average behaves when the quantity is changing at a constant rate. See the page Calculating with constant acceleration.)

Joe Redish 4/28/15


Article 335
Last Modified: June 2, 2021