Representing and Rearranging Physical Relationships

Hello, and welcome. This course builds the mathematical language needed to describe physical systems, then develops calculus-based mechanics, electricity and magnetism, and finally the foundations of relativity, quantum physics, and atomic/nuclear physics. We begin with a fundamental scientific habit: expressing one physical relationship in several equivalent forms.

In this lesson, you will represent a simple relationship using a formula, a table, and a graph. You will also rearrange a formula to make the quantity you need the subject. These are not separate skills: each representation reveals something the others hide.

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One physical relationship, three views

Suppose a train leaves a station and moves at approximately constant speed. We might want to describe how its distance from the station changes with time.

There are three useful representations:

1. A formula expresses the general rule.
2. A table shows particular input-output pairs.
3. A graph makes the overall pattern visible.

For constant-speed motion beginning at the station, the relationship is

$$d=vt$$

where:

• $d$ is distance from the station,
• $v$ is the train’s speed,
• $t$ is elapsed time.

If the speed is $2.0km/min$, the formula, written with its units made explicit, is

$$d=\left(2.0\frac{\mathrm{km}}{\min }\right)t$$

The formula says: for every minute that passes, the train is $2.0\mathrm{km}$ farther from the station.

A table gives several concrete cases of that rule:

Time, $t$ (min)	Distance, $d$ (km)
0	0
10	20
20	40
30	60
40	80
50	100
60	120
70	140

Every row satisfies the same formula. For instance, at $t=30\min$,

$$d=\left(2.0\frac{\mathrm{km}}{\min }\right)(30\min )=60\mathrm{km}$$

Notice what happens to the units: minutes cancel, leaving kilometres. Unit handling will become a central checking tool in the next lesson.

A graph displays these pairs as points. Conventionally, time is placed on the horizontal axis because it is the independent variable: we choose or observe a time, then determine the associated distance. Distance belongs on the vertical axis because it depends on the elapsed time.

The train data in the reading come from measurements, so the plotted points do not lie perfectly on one straight line. A line drawn close to the overall pattern is a trend line. It represents a simple model of the motion, rather than claiming every measurement is exact.

The trend line passes through $(0\min ,0\mathrm{km})$ and $(70\min ,140\mathrm{km})$. The train covers $140\mathrm{km}$ in $70\min$, consistent with a rate of $2.0km/min$. Thus the graph, the table, and the formula all encode the same approximate relationship.

A useful distinction:

• Data points report observations.
• A formula states a rule or model.
• A trend line shows the model visually.

A good model should agree reasonably with data within the limits of measurement uncertainty, but it need not pass through every point.

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Position can include a starting point

Distance from a station is often zero initially. But many physical situations begin somewhere other than the coordinate origin. For one-dimensional constant-velocity motion, the more general formula is

$$x=x_0+vt$$

Here, $x$ is the position at time $t$, and $x_0$ is the initial position: the position when $t=0$.

Consider a cart moving in the positive direction along a straight track. At the chosen starting time it is already $15\mathrm{m}$ from the origin, and it moves at constant velocity $1.2m/s$. Its model is

$$x=15\mathrm{m}+\left(1.2\frac{\mathrm{m}}{\mathrm{s}}\right)t$$

The formula produces this table:

Time, $t$ (s)	Position, $x$ (m)
0	15
5	21
10	27
15	33
20	39

For example, at $t=10\mathrm{s}$,

$$x=15\mathrm{m}+\left(1.2\frac{\mathrm{m}}{\mathrm{s}}\right)(10\mathrm{s})$$ $$x=15\mathrm{m}+12\mathrm{m}=27\mathrm{m}$$

A graph of this relationship is a straight line, but unlike the train model, it does not pass through the origin. At $t=0$, the cart is at $x=15\mathrm{m}$. That vertical-axis value is the graph’s initial position, also called its intercept.

The expression

$$x-x_0$$

is called the displacement, often written as $\Delta x$. Therefore,

$$\Delta x=vt$$

This does not mean that position and displacement are interchangeable. Position says where the object is relative to a chosen origin; displacement says how much its position changed between two times.

For now, read a position-time graph qualitatively:

• A horizontal line represents no change in position.
• A line rising as time increases represents motion in the chosen positive direction.
• A line falling as time increases represents motion in the negative direction.
• A steeper line represents a greater change in position per unit time.

In a later lesson, you will make this last statement precise by interpreting the slope as velocity.

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Rearranging a physical formula

Physics formulas are tools, not fixed instructions about which quantity must be calculated. The same relationship

$$x=x_0+vt$$

can answer different questions depending on what you know and what you need.

Suppose the cart starts at $x_0=15\mathrm{m}$, reaches $x=75\mathrm{m}$, and travels at $v=1.2m/s$. To find the elapsed time, make $t$ the subject.

Start with

$$x=x_0+vt$$

Subtract $x_0$ from both sides:

$$x-x_0=vt$$

Then divide both sides by $v$:

$$t=\frac{x-x_0}{v}$$

Substitute the physical values:

$$t=\frac{75\mathrm{m}-15\mathrm{m}}{1.2m/s}$$ $$t=\frac{60\mathrm{m}}{1.2m/s}=50\mathrm{s}$$

The algebra and the units tell the same story: metres divided by metres per second gives seconds.

From the same original formula, you can isolate any variable:

Quantity wanted	Rearranged form	Conditions
Position $x$	$x=x_0+vt$	—
Initial position $x_0$	$x_0=x-vt$	—
Velocity $v$	$v=\frac{x-x_0}{t}$	$t\ne 0$
Time $t$	$t=\frac{x-x_0}{v}$	$v\ne 0$

The governing principle of rearrangement is simple: whatever operation you perform on one side of an equation, perform on the other side as well.

Two common errors are worth preventing early:

• Treating $vt$ as though it were $v+t$. It means multiplication: velocity times time.
• Dividing only one term in a sum. For example, from $x=x_0+vt$, you cannot obtain $x/v=x_0+t$. Division by $v$ would have to apply to the entire right-hand side, including $x_0$.

After rearranging, check your result by substituting it into the original formula. With $t=50\mathrm{s}$,

$$x=15\mathrm{m}+\left(1.2\frac{\mathrm{m}}{\mathrm{s}}\right)(50\mathrm{s})$$ $$x=15\mathrm{m}+60\mathrm{m}=75\mathrm{m}$$

The result reproduces the known final position, so it is consistent.

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Moving confidently among words, formulas, tables, and graphs

When confronting a simple physical situation, use a short modelling routine.

1. Define the physical quantities and their units.
   Decide what is changing and choose clear symbols. For one-dimensional motion, $t$ commonly denotes time and $x$ position.

2. State the assumptions.
   The formula $x=x_0+vt$ assumes constant velocity over the time interval considered. It is not appropriate if the object repeatedly speeds up or slows down.

3. Write a formula before inserting numbers.
   This preserves the general relationship and reduces the chance of mixing up quantities.

4. Generate a small table.
   Choose several sensible values of the independent variable, calculate the dependent variable, and retain units in the headings.

5. Construct the graph carefully.
   Give it a descriptive title, label both axes with quantities and units, use a scale that includes the relevant values, and plot the table entries as ordered pairs.

6. Rearrange only after identifying the unknown.
   Ask which variable the problem actually requests, isolate that variable symbolically, then substitute numerical values.

This routine applies beyond motion. A spring’s extension may depend on applied force; electric current may depend on voltage; the energy of an object may depend on its speed. The physical meanings change, but the mathematical work of relating variables remains recognizably similar.

One final caution: a graph is not automatically a picture of an object’s path through space. A position-time graph has time on one axis and position on the other. It shows how one quantity changes with another, not the literal shape of the object’s trajectory.

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Wrap-up

You can now describe a simple physical relationship in three connected forms:

• A formula expresses the general model, such as $x=x_0+vt$.
• A table gives selected numerical cases that satisfy or test the model.
• A graph displays the relationship visually, with the independent variable usually on the horizontal axis and units on both axes.
• A formula can be rearranged systematically to solve for a chosen variable, while units and substitution provide valuable checks.

Next, you will strengthen this language by using dimensional analysis: testing whether a physical equation is even possible by checking the dimensions of its terms.