Position, Path Length and Displacement

Frame of reference and origin — the silent first step

NCERT opens the motion chapter with "Motion is change in position of an object with time." To speak of position, you must first plant a reference point in space — the origin. The full coordinate system (origin, axis direction, unit of length, clock) is called a frame of reference.

For motion in a straight line, the frame collapses to one axis. You pick a direction and call it positive; the opposite is negative. NCERT Summary point 1 states it plainly: "position to the right of the origin is taken as positive and to the left as negative." The choice is a convention, not a property of nature — but once chosen, it must be applied consistently to every quantity: position, displacement, velocity, acceleration, even $g$.

NIOS adds the licence the whole chapter operates under: "For 1-D motion, the directional aspect of the vector is taken care of by putting + and − signs and we do not have to use vector notation." You will not see arrows in 1D kinematics; you will see signs. They carry the same directional information.

The other baked-in approximation is the point object: "the size of the object is much smaller than the distance it moves." A train between stations qualifies; a tumbling beaker mid-fall does not. The idealisation is what lets a single number $x(t)$ describe the whole body.

Position as a signed coordinate

With origin and positive direction fixed, the position of a point object at any instant is a single real number $x$ on the axis. It is signed: $x = +5$ m means 5 m on the positive side, $x = -5$ m means 5 m on the negative side, $x = 0$ means at the origin. The unit is the metre.

Two consequences. First, position changes when the origin changes — slide the origin 3 m right and every position drops by 3. That is a relabelling, not a physical change. Second, the same motion can be described by infinitely many position-time graphs, one for each origin choice; they all share the same shape and differ only by a vertical shift.

NCERT Exercise 2.6 makes the dependence explicit: it instructs the student to "Choose the x = 0 m and t = 0 s to be the location and time of the ball at its highest point, vertically downward direction to be the positive direction" before asking for signs of position, velocity and acceleration. The origin and sign convention are part of the problem, not background.

Path length — a strict scalar

The path length (also called distance travelled, or total distance) of a body between an initial time $t_1$ and a final time $t_2$ is the total length of the trajectory it actually traces out in space. Three properties define it:

1. It is a scalar. There is no direction attached — only a non-negative number.
2. It is always non-negative. You can never travel a negative length of road, no matter which way you walk.
3. It never decreases with time. Path length only accumulates. Even if the object reverses direction, the new motion adds to the existing total — it does not subtract.

NIOS: "the total length of the path covered by a body is the distance travelled by it… distance is a scalar." NCERT Exercise 2.9(a) marks it as distinct from the magnitude of displacement.

Mental picture: an odometer ticking as an ant walks the trajectory. The odometer only counts forward. Whether the body moves east or west or reverses three times, the odometer accumulates. The final reading is the path length — path-dependent, so two routes between the same endpoints give two different path lengths even though their displacement is identical.

Displacement — the signed change in position

The displacement of a body between time $t_1$ and time $t_2$ is the change in its position:

where $x_1$ is the position at $t_1$ and $x_2$ is the position at $t_2$. Three properties:

1. It is fundamentally a vector. In one dimension, the direction reduces to a sign, so we treat $\Delta x$ as a signed scalar — but the sign carries the directional information.
2. It can be positive, zero or negative. Positive means the final position is on the positive side of the initial one; negative means the opposite; zero means the body returned to where it began (or never moved).
3. It is path-independent. Displacement depends only on the initial and final positions, not on the route between them. Whatever happens in between — reversals, pauses, zig-zags — is invisible to $\Delta x$.

The algebraic check: pick origin and positive direction, read $x_1$ and $x_2$, subtract. The sign is automatic. If the body moved further along the positive direction, $\Delta x > 0$; if further back, $\Delta x < 0$; if returned to start, $\Delta x = 0$. Because the origin contributes equally to $x_1$ and $x_2$, displacement is independent of the origin (though not of the positive direction).

Path length vs displacement — the comparison NEET tests

The most-tested idea in this subtopic is the inequality:

with equality only for unreversed straight-line motion. NCERT Exercise 2.9 asks exactly this and answers: "only when the motion is unidirectional, without retracing any part of the path."

Three NEET-style cases. (i) Train from A to B without reversal: path length $=$ |displacement|. (ii) NCERT Exercise 2.10 — walk 2.5 km to market and return: path length $= 5$ km, displacement $= 0$. (iii) NIOS Example 2.2 — one lap on a 300 m circular track: path length $= 300$ m, displacement $= 0$.

Worked examples — straight, reversal, round-trip

From displacement to velocity — a one-line bridge

Displacement is the raw material of velocity. NCERT and NIOS both define average velocity as $\bar v = \Delta x / \Delta t$. Because $\Delta x$ inherits a sign, average velocity does too. Average speed, by contrast, is total path length over time — non-negative by construction. The two coincide only for unreversed straight-line motion. The chain to remember is position → displacement → velocity. Get the sign wrong on the first link and every later answer silently inherits the error. The full velocity story belongs to the next subtopic.