Motion • Topic 1 of 3

Distance, Displacement & Uniform Motion

Everything around us is moving in some way — a bus on a Delhi flyover, a fan blade, the Earth around the Sun. An object is said to be in motion when its position changes with respect to a reference point (also called the origin or frame of reference). The same object can appear to be moving to one observer and at rest to another. A passenger sitting in a moving train is at rest relative to the seat, but in motion relative to a person standing on the platform. This is why we must always fix a reference point before describing motion.

To measure how much an object has moved, we use two different quantities. Distance is the total length of the actual path travelled by the object, regardless of direction. It is a scalar quantity — it has only magnitude, never a direction. Displacement is the shortest straight-line gap between the starting point and the final position, measured along with its direction. It is a vector quantity — it has both magnitude and direction. The SI unit of both is the metre (m).

A simple example makes the difference clear. If you walk 4 m east and then 3 m north, the distance covered is $4 + 3 = 7\,\text{m}$, but the displacement is the straight gap from start to finish, which is $\sqrt{4^2 + 3^2} = 5\,\text{m}$ towards the north-east. Distance is always positive and can never be less than the magnitude of displacement. Displacement can be positive, negative or even zero — if a runner completes one full lap of a circular track and returns to the start, the distance equals the track length but the displacement is zero.

Motion is classified by how the distance changes with time. In uniform motion, an object covers equal distances in equal intervals of time, no matter how small those intervals are. A car moving steadily at $60\,\text{km/h}$ on an empty highway is in uniform motion. In non-uniform motion, the object covers unequal distances in equal time intervals — like a car in city traffic that keeps speeding up and slowing down at signals. Most real-life motion is non-uniform.

Reference point: a fixed point chosen to describe whether and how an object moves.
Distance: total path length; scalar; always $\ge 0$.
Displacement: shortest start-to-end gap with direction; vector; can be zero.
Uniform motion: equal distances in equal time intervals.
Non-uniform motion: unequal distances in equal time intervals.

Solved Examples

Worked Example

A boy walks 6 m east, then 8 m north. Find the distance and the magnitude of displacement.

Solution

Distance = total path = $6 + 8 = 14\,\text{m}$.
The two legs are perpendicular, so displacement $= \sqrt{6^2 + 8^2}$.
$= \sqrt{36 + 64} = \sqrt{100} = 10\,\text{m}$.

Answer: Distance $= 14\,\text{m}$, Displacement $= 10\,\text{m}$.

Worked Example

A runner completes one full round of a circular track of radius $r = 35\,\text{m}$. Find the distance and displacement. (Take $\pi = \tfrac{22}{7}$.)

Solution

Distance for one full round = circumference $= 2\pi r$.
$= 2 \times \frac{22}{7} \times 35 = 220\,\text{m}$.
The runner returns to the start, so the start and end points are the same.
Displacement = shortest gap between same points $= 0$.

Answer: Distance $= 220\,\text{m}$, Displacement $= 0$.

Worked Example

An object moves from $x = 2\,\text{m}$ to $x = 10\,\text{m}$ along a straight line and then back to $x = 5\,\text{m}$. Find distance and displacement.

Solution

Forward path: $10 - 2 = 8\,\text{m}$. Return path: $10 - 5 = 5\,\text{m}$.
Distance = total path $= 8 + 5 = 13\,\text{m}$.
Displacement = final position $-$ initial position $= 5 - 2 = 3\,\text{m}$.
It is positive, so the net shift is in the direction of increasing $x$.

Answer: Distance $= 13\,\text{m}$, Displacement $= 3\,\text{m}$.

Worked Example

A car covers $15\,\text{m}$ in each second for $5$ seconds on a straight road. State the type of motion and the total distance.

Solution

Equal distance ($15\,\text{m}$) is covered in each equal interval (1 s).
By definition this is uniform motion.
Total distance $= 15 \times 5 = 75\,\text{m}$.

Answer: Uniform motion; total distance $= 75\,\text{m}$.

Worked Example

In successive seconds a body covers $2\,\text{m}, 4\,\text{m}, 6\,\text{m}, 8\,\text{m}$. Is the motion uniform or non-uniform? Justify.

Solution

Check the distances in equal time intervals of 1 s each.
They are $2, 4, 6, 8\,\text{m}$ — unequal and increasing.
Equal distances are not covered in equal times, so the motion is non-uniform.

Answer: Non-uniform motion (distances are unequal in equal time intervals).

Worked Example

A person walks $12\,\text{m}$ towards east and then $12\,\text{m}$ towards west. Find the distance and displacement.

Solution

Distance = total path $= 12 + 12 = 24\,\text{m}$.
East and west are opposite directions, so the second leg cancels the first.
Net shift from start $= 12 - 12 = 0$.

Answer: Distance $= 24\,\text{m}$, Displacement $= 0$.

Key Points

Motion means change of position with respect to a fixed reference point.
Distance is the total path length — a scalar, always $\ge 0$.
Displacement is the shortest start-to-end gap with direction — a vector, and can be zero.
Distance is always greater than or equal to the magnitude of displacement.
Uniform motion = equal distances in equal times; non-uniform = unequal distances in equal times.

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.Which of the following is a vector quantity?

Explanation: Displacement has both magnitude and direction, so it is a vector. Distance, speed and time are scalars.

Q2.An athlete runs once around a 400 m circular track and stops at the start. The displacement is:

Explanation: Start and end points coincide, so the shortest gap (displacement) is zero, even though distance is 400 m.

Q3.A body covers 5 m, 5 m, 5 m in each successive second. This is:

Explanation: Equal distances are covered in equal time intervals, which is the definition of uniform motion.

Q4.The SI unit of displacement is the:

Explanation: Displacement is a length, so its SI unit is the metre (m).

Practice more MCQs → 📝 Assignment · 30 marks