1.2 Position as a Function of Time

The GPS trace on your phone

Imagine you spent an afternoon walking through a city. Your phone recorded a GPS trace of the entire trip — a wiggly line on a map, annotated with timestamps every few seconds. That trace is a complete record of your walk. Someone looking at it can tell where you were at 2:15 PM, when you stopped for coffee, which direction you were heading at any moment.

Now suppose a friend calls and asks, "What did your walk look like?" You could describe it in words: I left the apartment, walked east for a few blocks, turned north, stopped at a cafe, then looped back home. That's a story — and it captures some of the truth. But it leaves out enormous amounts of detail. How fast were you moving on each block? Exactly when did you turn? How long was the coffee stop?

Here's the question that drives this section:

How would you communicate your entire walk to someone over the phone using only math?

Words are imprecise. A map is visual but hard to transmit. What we need is a function — a mathematical rule that assigns a position to every moment in time. That function would contain everything. And that is exactly what physics builds.

Prediction: reading motion from a graph

Before we formalize anything, let's see if you can already read motion from a picture.

Prediction exercise

Below are two position-vs-time graphs, each showing one person's motion along a straight road over 10 seconds.

[Interactive: Two side-by-side position-vs-time graphs. Graph A shows a straight line rising steeply from $x = 0$ to $x = 30$ m over 10 s. Graph B shows a line that rises from $x = 0$ to $x = 15$ m over 5 s, then falls back to $x = 0$ over the next 5 s. Students must answer two questions before the explanation unlocks: (1) "Which person is moving faster during the first 5 seconds?" with options A / B / Same speed. (2) "Which person turned around at some point?" with options A / B / Neither.]

Commit to your answers before reading on.

After you commit:

Person A covers 30 meters in 10 seconds along a straight line — a steady 3 m/s. During the first 5 seconds, A is at position 15 m (halfway along). Person B also reaches 15 m at $t = 5$ s — so during the first 5 seconds, they move at the same speed. Many students pick A here because A ends up farther away, but the slopes of both graphs are identical over that interval.

The real difference is what happens next. After $t = 5$ s, Person B's graph slopes downward. A downward slope on a position-time graph means the person is moving backward — returning toward the starting point. Person B turned around at $t = 5$ s.

The key insight: the slope of a position-time graph tells you about speed and direction. A steeper slope means faster motion. A positive slope means motion in the positive direction; a negative slope means motion the other way. A flat graph means the person is standing still.

You just read physical meaning from a mathematical picture. That is the core skill of this section.

What does it mean to describe motion by a function?

In your calculus courses, you worked with functions like $f(x) = x^2 - 3x + 1$. You gave the function an input, it returned an output. The input was usually called $x$, and the output was some number.

In physics, we do exactly the same thing, but the input and output have physical meaning:

• Input: a moment in time, $t$
• Output: a location in space, $x$

When we write $x(t) = 2t + 1$, we mean: at time $t$, the object is at position $2t + 1$. Plug in $t = 0$ and you get $x = 1$ — the object starts at position 1. Plug in $t = 3$ and you get $x = 7$ — three seconds later, it's at position 7.

The function $x(t)$ is a complete instruction set for the motion. Every question about where the object is, and when, has an answer — just evaluate the function.

$$x(t) = \text{position at time } t$$

This is a small but profound shift in thinking. Instead of describing motion in sentences ("the ball rolls to the right and speeds up"), we capture it in a rule that works for every moment at once. The function doesn't just tell a story — it tells the whole story, with every detail pinned down.

Exploration: build a motion from a function

Time to get your hands on this idea.

[Interactive: Function-to-motion explorer. A text input field where students type a function of $t$ (e.g., $x(t) = 2t + 1$). Below the input, three synchronized views appear:

1. A number line with a dot that moves in real time as $t$ advances from 0 to 6 seconds.
2. A position-vs-time graph that draws itself as the animation plays.
3. A data table showing $t$ and $x(t)$ at integer times: $t = 0, 1, 2, 3, 4, 5, 6$.

All three views update live when the student changes the function. A play/pause button controls the animation. The input field pre-loads with $x(t) = 2t + 1$.]

Work through these guided challenges:

Challenge 1: The default function is $x(t) = 2t + 1$. Watch the dot. Where does it start? Which direction does it go? Is it speeding up, slowing down, or moving at a constant pace?

Challenge 2: Change the function to $x(t) = 5$. What happens? Watch both the dot and the graph. What does a constant function mean physically?

Challenge 3: Make the dot start at position 5 and move to the left. (Hint: moving left means position is decreasing.) Try $x(t) = 5 - t$. What does the graph look like?

Challenge 4: Try $x(t) = t^2$. How is this motion different from the straight-line motions? Watch the dot carefully — is it moving at a constant pace?

Challenge 5: Can you write a function where the dot starts at $x = 3$, moves to the right, and then comes back? (Hint: think about what kind of function rises and then falls.)

Take a few minutes with this. Change the function, watch the dot, check the graph, look at the table. The goal is to build a gut feeling for how the shape of $x(t)$ controls the motion of the object.

Concept reveal: the position function as a complete model

Here's what you just discovered through exploration:

The position function $x(t)$ is a complete mathematical model of one-dimensional motion. Once you have it, you can answer any question about the motion:

• Where is the object at $t = 4$? Evaluate $x(4)$.
• When is the object at position 10? Solve $x(t) = 10$ for $t$.
• Does the object ever return to its starting point? Check whether $x(t) = x(0)$ has a solution for $t > 0$.
• Is the object speeding up or slowing down? (We'll formalize this in Section 1.5, but you can already see it in the graph's curvature.)

In more than one dimension, we use a vector-valued function $\vec{r}(t) = \langle x(t),\, y(t) \rangle$ (or $\langle x(t),\, y(t),\, z(t) \rangle$ in 3D). Each component tells you position along one axis. The idea is the same: one function, complete information.

Think of $x(t)$ as a set of instructions you could hand to a robot. If the robot evaluates $x(t)$ at every instant and moves to the corresponding position, it would reproduce the motion perfectly. The function is the motion, encoded in mathematics.

Four representations of the same motion

A position function can be expressed in multiple ways, and each one reveals something different. Let's look at the same motion through four lenses.

Consider an object whose position is given by $x(t) = 4t - t^2$ (measured in meters, with time in seconds).

The equation

$$x(t) = 4t - t^2$$

From the equation, you can spot structure: this is a downward-opening parabola. The object starts at $x(0) = 0$, and the position becomes zero again when $4t - t^2 = 0$, i.e., $t(4 - t) = 0$, so at $t = 0$ and $t = 4$ s. The equation is compact and general — it works for every value of $t$ at once.

The data table

The table shows specific values. You can see that the position increases, reaches a maximum, and then decreases — the object goes out and comes back. The table is concrete and easy to read, but it only shows the moments you chose to include. What happens at $t = 1.5$? The table doesn't say directly, though you could interpolate.

The graph

The graph shows the shape of the motion at a glance. You can immediately see when the object is moving forward (graph rising), when it's moving backward (graph falling), and when it momentarily stops (the peak). The graph captures the qualitative story — the overall pattern — better than any other representation.

The animation

The animation shows what the motion looks like — the physical reality that all the math describes. Notice how the dot slows down as it approaches the peak and speeds up again on the way back. The synchronized graph-and-animation view lets you connect the mathematical picture (the curve) to the physical picture (the moving dot).

Pause and reflect: Each representation tells the same story. The equation is the most general. The table is the most concrete. The graph is the best for seeing patterns at a glance. The animation is the closest to physical reality. Physics fluency means being comfortable with all four.

Variation: what changes, what stays the same?

Let's build your intuition by comparing three related functions. In each case, only one thing changes.

[Interactive: Synchronized comparison panel. Three position functions are displayed side by side, each with its equation, graph, and animated dot on a shared number line. Students can play all three animations simultaneously or one at a time.

• Function A: $x(t) = 2t$
• Function B: $x(t) = 2t + 3$
• Function C: $x(t) = 5t$

All three share the same time axis (0 to 4 s) and position axis (0 to 25 m).]

Comparing A and B: Both graphs are straight lines with the same slope. Function B is just shifted upward by 3. What does this mean physically?

The "+3" changes the starting position but not the speed. Both objects move at the same rate; B just starts 3 meters ahead of A. The slope is the same, so the speed is the same.

Comparing A and C: Both start at $x = 0$ (same starting position), but C has a steeper slope. What does steeper mean physically?

A steeper slope on the position-time graph means faster motion. Object C covers more distance in the same time. The coefficient in front of $t$ controls the speed.

What changed? What stayed the same?

Going from A to B: starting position changed, speed stayed the same. Going from A to C: speed changed, starting position stayed the same.

This pattern — varying one feature while holding everything else constant — is how you build reliable intuition. When you see a new function, you can mentally decompose it: "What controls the starting position? What controls the speed? What controls whether the object turns around?"

Practice layers

Layer 1: Concrete — reading from a graph

1. "What is the position at $t = 2$ s?" (Answer: 6 m)
2. "During which time interval is the object not moving?" (Answer: $t = 3$ to $t = 5$ s)
3. "At what time(s) is the object at position $x = 5$ m?" (Answer: approximately $t = 1.5$ s and $t = 6.5$ s)
4. "During which interval is the object moving in the negative direction?" (Answer: $t = 5$ to $t = 8$ s)

Layer 2: Pattern — matching equations to motions

[Interactive: Matching exercise. Four short animations play on the left (labeled P, Q, R, S), each showing a dot moving on a number line for 4 seconds. On the right, four equations are listed:

• $x(t) = 3t$
• $x(t) = 12 - 3t$
• $x(t) = 6$
• $x(t) = t^2$

Students drag equations to match them with the correct animation. Feedback after each match, with brief explanation of why the equation produces that motion.]

The key to matching: look at the starting position ($x$ when $t = 0$), the direction of motion (is $x$ increasing or decreasing?), and whether the speed is constant or changing.

Layer 3: Structure — same endpoints, different journeys

Consider two position functions, both describing motion from $t = 0$ to $t = 4$ s:

$$x_1(t) = 3t, \qquad x_2(t) = \tfrac{3}{16}\,t^3$$

Compute $x_1(0)$ and $x_2(0)$. Then compute $x_1(4)$ and $x_2(4)$. Do these objects start and end at the same position?

Both functions give $x(0) = 0$ and $x(4) = 12$ m. Same starting position, same final position.

But do they describe the same motion?

Build the table and see:

Not at all the same motion. Object 1 moves at a steady pace, covering 3 meters every second. Object 2 barely crawls at the start and then rushes to catch up at the end — most of its distance is covered in the final two seconds.

The lesson: Knowing where something starts and ends is not enough to describe its motion. The position function tells you how the object got there — the entire journey, not just the departure and arrival. Two different functions can share endpoints but describe completely different physical experiences.

Layer 4: Transfer — same math, different domain

A stock is priced at $P(t) = 50 + 8t - t^2$ dollars, where $t$ is measured in months after January.

1. What was the price in January ($t = 0$)?
2. When did the price peak?
3. When did the price return to its January value?

This is exactly the same mathematics as a position function. The "position" is now price. The "time" is still time. The graph has the same shape (inverted parabola). The skills you built for interpreting $x(t)$ transfer directly.

Answers: 1. $P(0) = 50$ dollars. 2. The peak occurs at $t = 4$ months (completing the square or using $t = -b/(2a) = -8/(2 \cdot (-1)) = 4$), where $P(4) = 50 + 32 - 16 = 66$ dollars. 3. Solve $P(t) = 50$: $8t - t^2 = 0$, so $t(8 - t) = 0$, giving $t = 0$ or $t = 8$. The price returns to \$50 after 8 months.

The position function is not just a physics tool — it's a mathematical lens for anything that changes over time.

Connection to what you already know

You already know functions from calculus. You know how to evaluate them, graph them, find their zeros, and identify their maxima. Everything in this section uses those same skills.

What's new is the interpretation. In calculus, $f(x)$ was an abstract relationship between numbers. Now $x(t)$ carries physical meaning: the input is a clock reading, the output is a location in space. The slope of the graph isn't just "rise over run" — it tells you how fast the object is moving. A maximum isn't just a critical point — it's the moment the object turns around.

In Section 1.1, you learned that physics simplifies reality through models. The position function is your first real model: a mathematical object that stands in for a physical process. In the next sections, you'll see how choosing a reference frame (Section 1.3) affects how you write this function, and how calculus extracts velocity (Section 1.5) and acceleration (Section 1.6) from it.

Everything in this course flows from $x(t)$. It is the foundation.

Reflection

Take a moment to think about the four representations you worked with in this section: equation, graph, data table, and animation.

• Which one feels most natural to you right now? Which one do you reach for first when trying to understand a motion?
• Which one feels least natural? Which one would you avoid if you could?

The representation that feels least natural is the one worth practicing most. Over the coming sections, we'll use all four repeatedly, and the goal is to become fluent in translating between them.

Summary

• The position function $x(t)$ assigns a location to every moment in time. It is a complete mathematical model of one-dimensional motion.
• In two or three dimensions, the position function becomes a vector: $\vec{r}(t) = \langle x(t), y(t) \rangle$ or $\langle x(t), y(t), z(t) \rangle$.
• A position function can be represented as an equation, a graph, a data table, or an animation. Each reveals different features of the motion.
• On a position-vs-time graph: the slope encodes speed and direction, a flat section means no motion, and a peak or valley means the object reverses direction.
• Two functions can share the same start and end positions but describe completely different motions. The function captures the entire journey, not just the endpoints.
• The mathematics of position functions is the same as the function analysis you know from calculus — what's new is the physical interpretation.