Describing Motion — Print · Grade 12 Science

Describing Motion

Learning goals

• I can describe an object's position relative to a chosen origin and reference frame.
• I can distinguish distance from displacement, and scalars from vectors, and calculate a resultant displacement.
• I can explain how the same motion looks different from different frames of reference.

Readings for this unit

• One ball, two component stories and a shared clock

Before you can say how something moves, you have to agree on where it is, and that always starts with a choice. You pick an origin — a reference point you're calling zero — and then an object's position is just its location measured from that origin, complete with a sign that tells you which side it's on. A hiker 5 km east of the trailhead is at position +5 km; a diver 5 km west is at −5 km. Physics doesn't hand you the origin; you choose it to make the problem simple, and every number that follows is measured against it.

Once positions are pinned down, two very different questions about a trip come apart. Distance is the total path length actually traveled — every step, every detour, added up — and it never carries a direction or a sign. Displacement is the straight-line change in position from start to finish, written Δx = x_final − x_initial, and it does carry direction. Walk 40 m east and then 30 m west and you've covered a distance of 70 m, but your displacement is only 10 m east. Run one full lap of a track and return to the start, and you've logged hundreds of meters of distance for a displacement of exactly zero.

That contrast is really the difference between two families of quantities. A scalar has size only — distance, speed, time, temperature, and mass are all scalars, fully described by a number and a unit. A vector has both size and direction — displacement, velocity, and acceleration are vectors, and leaving out the direction throws away half the information. This is why physicists are so fussy about arrows and signs: '10 m' answers a different question than '10 m east,' and in two dimensions the direction can be the whole point.

Vectors also combine differently than plain numbers, because directions can partly cancel or reinforce. When two displacements lie along the same line you simply add or subtract them, but when they meet at a right angle you combine them with the Pythagorean theorem. A drone that flies 40 m east and then 30 m north hasn't ended up 70 m from home — it's √(40² + 30²) = √2500 = 50 m away, along a diagonal. The path length (distance) and the straight-line result (displacement magnitude) genuinely differ, and only careful vector arithmetic gets the second one right.

Every one of these measurements secretly depends on a frame of reference — the origin and coordinate system you're measuring from. Sit still reading on a moving train and, in the train's frame, your velocity is zero; in the platform's frame you're rushing past at 100 km/h; in the Sun's frame you're hurtling through space at tens of kilometers per second. None of these is 'the real' answer, because there is no absolute, universal resting point in the universe. Motion is always relative to a chosen frame, and two observers in different frames can honestly disagree about how fast — or even whether — something is moving.

Getting fluent with these ideas is what makes the rest of kinematics possible. Position gives you the 'where,' displacement and distance separate the net change from the total journey, the scalar–vector split keeps direction in the accounting, and the frame of reference reminds you to state whose point of view you're using. Air-traffic controllers, GPS receivers, and self-driving cars all run on exactly this framework — tracking positions as coordinates and displacements as vectors, always relative to a clearly defined reference frame.

Model check: Distance traveled and displacement can differ even on a straight line. Distance 8 m; displacement+2 m with east positive.

Displacement

Δx = x_final − x_initial

x_final and x_initial are positions along a chosen axis (each with a sign); Δx is a vector, and its sign gives the direction. Distance, in contrast, is the total path length and is always positive.

Resultant of perpendicular displacements

d = √(Δx² + Δy²)

when two displacement legs meet at a right angle (e.g. east then north), combine them with the Pythagorean theorem; d is the straight-line displacement magnitude.

Worked examples

Avoid these mistakes

• Distance traveled and displacement can differ even on a straight line.

Words to know

position

An object's location relative to a chosen reference point (the origin), usually given as a coordinate with a sign for direction.

distance

The total length of the path an object travels; a scalar, always positive, with no direction.

displacement

The straight-line change in position from start to finish, with direction: Δx = x_final − x_initial. A vector.

scalar

A quantity with magnitude only, such as distance, speed, time, or mass.

vector

A quantity with both magnitude and direction, such as displacement, velocity, or acceleration.

frame of reference

The point of view — a chosen origin and coordinate system — from which position and motion are measured.

There is no such thing as absolute rest

It feels obvious that the ground beneath you is 'not moving,' but physics insists otherwise. Right now you are spinning with Earth's surface at hundreds of meters per second, orbiting the Sun at about 30 km/s, and drifting with the whole solar system through the galaxy faster still. Which of these is your 'true' speed? The answer is that there isn't one. Galileo's principle of relativity says the laws of motion are the same in every uniformly moving frame, so no experiment can single out an absolute resting point. This is why you can pour a drink on a smoothly cruising airplane exactly as you would at home — inside that frame, everything behaves as if you were at rest. Motion is only ever defined relative to a chosen frame of reference.

Why displacement can be smaller than distance — and never larger

Distance is the length of the actual path; displacement is the straight-line shortcut from start to finish. Since the straight line between two points is the shortest route, displacement can never exceed distance, and the two are equal only when the motion is a straight line that never reverses. The gap between them is a measure of how much the path wandered or doubled back: a 400 m lap has 400 m of distance but zero displacement, while a diagonal walk of 40 m east and 30 m north covers 70 m of path for just 50 m of displacement. Whenever a problem gives you a winding route, that gap is the first clue that distance and displacement will disagree.

Vectors: adding tip-to-tail and breaking into components

Vectors don't add like ordinary numbers because directions can reinforce or cancel. To add them graphically you place them tip-to-tail — the start of the second at the end of the first — and draw a single arrow from the very beginning to the very end; that arrow is the resultant. The reverse trick is even more powerful: any vector can be broken into perpendicular components, usually one horizontal and one vertical, using v_x = v·cosθ and v_y = v·sinθ. This 'resolve into components' move is the workhorse of the entire unit — it's exactly how a diagonal displacement becomes a clean right triangle, and how an angled launch becomes two independent motions in the projectile lesson.

From Descartes' grid to GPS coordinates

The idea of pinning a position to numbers on axes comes from René Descartes in the 1600 s, whose coordinate grid let geometry be written as algebra — which is why we still call them Cartesian coordinates. That same idea now runs the modern world: a GPS receiver finds your position by timing signals from at least four satellites, each broadcasting where it is and when, and solving for the one point in three-dimensional coordinates consistent with all of them. The receiver reports latitude, longitude, and altitude — a position vector relative to Earth's center — and tracks how that vector changes to compute your displacement and velocity. Every 'you are here' dot is a physics coordinate problem solved dozens of times a second.

In the real world

A GPS receiver reports your position as coordinates relative to Earth's center and computes your displacement as a vector, which is how a navigation app knows both how far you are from your destination and in which direction to send you.

Exit check

Walk 5 m east then 3 m west. Give distance and displacement.