Position, displacement, average velocity

Before we talk about “how fast” or “how much acceleration”, we must be able to say exactly “where”. This section defines the three foundations of all kinematics: position, displacement, and average velocity. Nail these and the rest of the chapter follows smoothly.

Reference frame

No position is meaningful without a reference. “The car is 5 meters” says nothing — 5 meters from what? So we start by choosing a reference frame:

1. An origin (origin) — a point in space we call “zero”
2. A directed axis (axis) — a line marked with an arrow to indicate the positive direction
3. A unit of measurement — usually m in SI

Example: we want to describe a cyclist moving down a street. Set the origin at “Shahida intersection”, the positive axis pointing north, and the unit m. Now “cyclist at x = 200 m” is meaningful: 200 meters north of Shahida intersection.

Important: the choice of origin and axis is arbitrary — every observer can pick differently, and physics gives the same laws. This is one of the deepest principles of physics (the principle of Galilean relativity), but for now we just use it.

Position x(t)

The position of an object at time t is written x(t). This is a signed number:

• If the object is in the positive direction of the axis: x > 0
• If in the negative direction: x < 0
• At origin: x = 0

Cyclist example:
– x(0) = 200 m — at time zero, 200 m north of origin
– x(60) = 800 m — after 60 s, 800 m north
– x(120) = 500 m — after 120 s, 500 m north (turned back a bit)
– x(180) = −100 m — after 180 s, 100 m south (negative sign!)

Displacement Δx

Displacement is the change of position between two times:

$$
\Delta x = x_2 – x_1 = x(t_2) – x(t_1)
$$

Three key properties:
– Vector (in 1D: signed). Can be positive, negative, or zero
– Depends only on the two endpoints — not on the path between them
– Its sign indicates direction — positive means motion in the positive direction of the axis

Cyclist example:

$$
\Delta x_{[0,60]} = 800 – 200 = +600\ \mathrm{m}
$$

$$
\Delta x_{[60,120]} = 500 – 800 = -300\ \mathrm{m}
$$

$$
\Delta x_{[0,180]} = -100 – 200 = -300\ \mathrm{m}
$$

Note the total [0, 180] displacement is not zero — because the endpoint differs from the start.

Distance traveled

Distance is the total path length traveled — always positive, regardless of direction.

For the cyclist in [0, 180]:
– [0, 60]: 200 → 800 = went 600 m north
– [60, 120]: 800 → 500 = returned 300 m south
– [120, 180]: 500 → −100 = went 600 m further south

Total distance = 600 + 300 + 600 = 1500 m
Total displacement = −100 − 200 = −300 m

This distinction matters: distance ≠ magnitude of displacement. They only match when the motion is without a change of direction.

Everyday example: swim a full length across the pool and back to your starting spot — distance = 50 m + 50 m = 100 m, displacement = 0. Both are correct, they just mean different things.

Average velocity

Average velocity is the average rate of displacement over time:

$$
v_{\mathrm{avg}} = \frac{\Delta x}{\Delta t} = \frac{x_2 – x_1}{t_2 – t_1}
$$

• Vector (signed). Its sign indicates the average direction
• SI unit: m/s
• Can be zero — if start and end coincide (like the pool example)

For the cyclist in [0, 60]:

$$
v_{\mathrm{avg}} = \frac{800 – 200}{60 – 0} = \frac{600}{60} = +10\ \mathrm{m/s}
$$

In [60, 120]:

$$
v_{\mathrm{avg}} = \frac{500 – 800}{120 – 60} = \frac{-300}{60} = -5\ \mathrm{m/s}
$$

Negative sign means motion in the south direction (opposite to the positive axis).

Average speed

Average speed is not the same as average velocity:

$$
s_{\mathrm{avg}} = \frac{\text{total distance}}{\Delta t}
$$

• Always positive (distance is positive, time is positive)
• Different meaning from average velocity — this is “on average, how fast were you going”, not “how fast did you displace”

For the cyclist in all of [0, 180]:

$$
s_{\mathrm{avg}} = \frac{1500}{180} = 8.33\ \mathrm{m/s}
$$

$$
v_{\mathrm{avg}} = \frac{-300}{180} = -1.67\ \mathrm{m/s}
$$

Very different! Average speed 8.33 m/s (total distance / time), but average velocity −1.67 m/s (because the endpoint is south of the origin).

Everyday usage: when you say “we averaged 60 km/h”, you usually mean average speed, not average velocity.

Graphical interpretation

On an x-t plot:
– Position = height of the point at each time t
– Displacement Δx between two points = vertical gap
– Average velocity = slope of the straight line connecting the two points:

$$
v_{\mathrm{avg}} = \frac{\Delta x}{\Delta t} = \text{slope of chord}
$$

• Positive slope → motion in the positive direction
• Negative slope → motion in the negative direction
• Zero slope → stopped (or a back-and-forth that canceled out)

Note: a straight line has constant slope → average velocity is the same over any interval. Meaning: the instantaneous velocity is also constant (as we’ll see in §2.2).

A few notes and common mistakes

1. The sign means direction, not magnitude.
v = −20 m/s is larger in magnitude than v = +5 m/s, but its direction is opposite. Be careful in calculations.

2. Don’t confuse velocity and speed.
– Speed = |velocity| when motion is one-directional
– But average velocity and average speed can differ hugely (as in the cyclist example)

3. Time always moves forward positively.
Δt = t_2 − t_1 > 0 — if you flip start and end, Δt < 0 and the sign of velocity flips too. Convention: t_1 < t_2.

4. The choice of origin doesn’t affect physics.
If you shift the origin by 100 m, every x value shifts by 100 m, but Δx stays the same. Physics depends on Δx, not on x alone.

5. Units.
Velocity in SI: m/s. Common in everyday life: km/h. Conversion (§1.9):

$$
1\ \mathrm{km/h} = \frac{1000\,\mathrm{m}}{3600\,\mathrm{s}} = 0.278\ \mathrm{m/s}
$$

Or inversely: 1 m/s = 3.6 km/h. Memorize these two — you’ll use them constantly.

What you should be able to do

After this section, you should be able to:

• For a given scenario, choose a reference frame (origin + axis + positive direction)
• Write the position of an object as x(t) with the correct sign
• Compute the displacement Δx between any two times, respecting sign
• Distinguish displacement from distance
• Compute average velocity and average speed, and know why they can differ dramatically
• Interpret a chord’s slope on an x-t plot as the average velocity
• Convert fluently between km/h and m/s

Preview of §2.2

If an object moves with a varying velocity, the “average velocity” over large intervals doesn’t tell you much — just “roughly how fast”. For precision, we need the “velocity at an instant“. There we take the limit (Δt → 0) and arrive at the derivative:

$$
v(t) = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}
$$

That is the topic of §2.2.

📚 See also: Halliday Vol 1, Ch 2, §2.1 — Position, Displacement, and Average Velocity.
📖 Open reference: OpenStax University Physics Vol 1 — Chapter 3.1: Position, Displacement, and Average Velocity.
📖 Feynman Lectures Vol I — Ch 8: Motion (an excellent introduction to kinematics from a physicist’s angle).