In physics, distance and displacement are two fundamental quantities used to describe motion in physics. This quiz tests your understanding of both concepts through 15 worked questions, each with a detailed step-by-step solution. Try to answer each question yourself before revealing the solution.
Distance and displacement are the only concepts tested in this quiz, but you should already be familiar with right-angle triangles and the Pythagorean theorem, since several solutions use it to calculate displacement.
Key Definitions
- Distance is the total path length covered by an object, irrespective of direction. It is a scalar quantity and can never decrease or be negative.
- Displacement is the shortest straight-line distance between an object's initial and final position, along with the direction from initial to final position. It is a vector quantity and can be positive, negative, or zero.
- For any motion, distance ≥ displacement. The two are equal only when the motion is along a straight line without any change in direction.
Distance and Displacement Practice Questions
Question 1: Displacement and the choice of origin
A particle travels along a curved path between two points $P$ and $Q$ as shown:
The displacement of the particle does not depend on:
- the location of $P$ and $Q$
- the choice of origin
- the direction of $Q$ from $P$
- the length of the line joining $P$ and $Q$
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Answer: (b) the choice of origin
Displacement = position vector of the final position − position vector of the initial position. This means displacement depends only on the locations of $P$ and $Q$, and is completely independent of where the origin is placed.
The direction of the displacement is from $P$ to $Q$, and its magnitude equals the length of the straight line joining the initial and final positions.
Question 2: Does zero displacement mean zero distance?
The displacement of a moving object in a given interval of time is zero. Would the distance traveled by the object also be zero?
- Yes
- No
- Depends on the situation
- None of the above
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Answer: (b) No
It is not necessary for distance to be zero. When a body returns to its starting point after traveling along a path, the displacement is zero, but the distance traveled is not zero — it equals the total length of the path taken.
Question 3: Total displacement after walking in four directions
What is the total displacement of a student who walks 4 blocks east, 2 blocks north, 3 blocks west, and then 2 blocks south?
- 2 blocks east
- 2 blocks west
- 1 block west
- 1 block east
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Answer: (d) 1 block east
Displacement is defined as the shortest straight-line distance between the starting and ending points, together with its direction.
The student walks 4 blocks east, then 2 blocks north, then 3 blocks west, and finally 2 blocks south. The north and south movements cancel out (2 blocks each), leaving a net east-west movement of $4 - 3 = 1$ block east.
So the total displacement of the student is 1 block east.
Question 4: Displacement and distance on three-quarters of a circle
A body moves on three-quarters of a circle of radius $r$. The displacement and distance traveled by it are:
- displacement $= r$, distance $= 3r$
- displacement $= \sqrt{2}\,r$, distance $= \dfrac{3\pi r}{2}$
- distance $= 2r$, displacement $= \dfrac{3\pi r}{2}$
- displacement $= 0$, distance $= \dfrac{3\pi r}{2}$
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Answer: (b) displacement $= \sqrt{2}\,r$, distance $= \dfrac{3\pi r}{2}$
The circumference of the full circle is $2\pi r$. Three-quarters of that circumference is:
$\text{Distance} = \dfrac{3}{4} \times 2\pi r = \dfrac{3\pi r}{2}$
Distance is simply the total path covered by the body, so this same value applies here.
Displacement is the shortest straight-line distance between the initial and final position ($AB$ in the figure). Since the initial and final radii are perpendicular to each other (three-quarters of a circle leaves a 90° angle between them):
$\text{Displacement} = \sqrt{OA^2 + OB^2} = \sqrt{r^2 + r^2} = \sqrt{2}\,r$
Question 5: Which quantity is not a vector?
Which of the following is not a vector?
- Displacement
- Distance
- Acceleration
- Velocity
Show Answer
Answer: (b) Distance
Distance has only magnitude and no direction, so it is a scalar quantity — unlike displacement, acceleration, and velocity, which are all vectors.
Question 6: SI unit of displacement
The SI unit of displacement is:
- Meter
- Centimeter
- Kilometer
- None of these
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Answer: (a) Meter
The metre (m) is the SI base unit used for displacement, since displacement is a length measurement in the International System of Units.
Question 7: Relationship between distance and displacement
For any object in motion, distance is always ……………… displacement.
- less than or equal to
- greater than or equal to
- equal to
- less than
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Answer: (b) greater than or equal to
Since displacement is the shortest possible path between two points and distance is the actual path covered, distance is always greater than or equal to displacement. The two are equal only for straight-line motion in one direction.
Question 8: Identifying distance vs. displacement from a description
The motion of a car in the northeast direction for 100 miles tells us about:
- Distance
- Displacement
- Speed
- Velocity
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Answer: (b) Displacement
This is displacement because both a magnitude (100 miles) and a direction (northeast) are specified. A statement of distance alone would give only the magnitude, with no direction.
Question 9: Is distance a scalar or vector quantity?
Distance is a …………. quantity.
- Scalar
- Vector
- Negative
- None of these
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Answer: (a) Scalar
Since distance has magnitude but no direction, it is classified as a scalar quantity.
Question 10: Possible values of displacement
Which of the following values can displacement take?
- Positive
- Negative
- Zero
- All of these
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Answer: (d) All of these
Because displacement is a vector, it can be positive, negative, or zero depending on the chosen reference direction and the object's path of motion.
Question 11: Displacement after three separate movements
A person moves 30 m north, 20 m east, and finally $30\sqrt{2}\,m$ in the south-west direction. The displacement of the person from the origin will be:
- 10 m along North
- 10 m along South
- 10 m along West
- Zero
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Answer: (c) 10 m along West
Given:
- Distance moved north $= 30\,m$
- Distance moved east $= 20\,m$
- Distance moved south-west $= 30\sqrt{2}\,m$
In the right-angled triangle $\triangle ECD$, applying the Pythagorean theorem:
$EC^2 = ED^2 + CD^2 \;\Rightarrow\; ED^2 = EC^2 - CD^2 = (30\sqrt{2})^2 - (30)^2 = 1800 - 900 = 900 \;\Rightarrow\; ED = 30\,m$
The displacement of the person from the origin is:
$AE = ED - AD = 30 - 20 = 10\,m$
So the displacement is 10 m along the west.
Question 12: Displacement from a position-time equation
A particle is moving along the x-axis whose position varies with time according to $x = -3t + t^3$, where $x$ is in meters and $t$ is in seconds. The displacement of the particle from $t = 1\,s$ to $t = 3\,s$ is:
- $+16\,m$
- $-16\,m$
- $+20\,m$
- $-20\,m$
Show Answer
Answer: (c) $+20\,m$
Using $x = -3t + t^3$:
At $t = 1\,s$: $x = -3(1) + (1)^3 = -2\,m$
At $t = 3\,s$: $x = -3(3) + (3)^3 = -9 + 27 = 18\,m$
Displacement $= x_f - x_i = 18 - (-2) = +20\,m$
Question 13: Resultant displacement and total distance for a delivery route
A girl delivering newspapers covers her route by traveling 3.00 blocks west, 4.00 blocks north, and then 6.00 blocks east.
(i) What is her resultant displacement?
- 13.00 blocks at 51.1° counterclockwise of East
- 5.00 blocks at 53.1° clockwise of East
- 5.00 blocks at 53.1° counterclockwise of East
- 13.00 blocks at 51.1° counterclockwise of East
(ii) What is the total distance she travels?
- 13.00 blocks
- 5.00 blocks
- 5.00 blocks
- 13.00 blocks
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Answer: (i) (c) 5.00 blocks at 53.1° counterclockwise of East | (ii) (d) 13.00 blocks
Net displacement along the x-axis (east-west): $-3 + 0 + 6 = +3$ blocks
Net displacement along the y-axis (north-south): $0 + 4 + 0 = +4$ blocks
By the Pythagorean theorem, the resultant displacement is:
$d = \sqrt{4^2 + 3^2} = \sqrt{25} = 5.00$ blocks
The angle the resultant makes with the eastward x-axis:
$\theta = \tan^{-1}\left(\dfrac{4}{3}\right) = \tan^{-1}(1.33) = 53.1°$
The total distance traveled is simply the sum of all segments: $3.00 + 4.00 + 6.00 = 13.00$ blocks.
Question 14: Distance and displacement on a half circle
Distance and displacement of a body when it moves through half a circle of radius $r$ is:
- $r, r$
- $\pi r, 2r$
- $2\pi r, r$
- $\pi r, 0$
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Answer: (b) $\pi r, 2r$
Distance is half the circumference of the circle: $\text{Distance} = \dfrac{1}{2}(2\pi r) = \pi r$.
Displacement is the straight-line distance between the start and end points, which for a half circle is the diameter: $\text{Displacement} = 2r$.
Question 15: Distance and displacement for a boat crossing a river
A boat sailing through a river moved eastward for 5 km, then crossed the river by moving 3 km southward. On reaching the other side, it moved westward through 1 km and reached the jetty. The distance covered and displacement of the boat respectively are:
- 9 km, 4 km
- 5 km, 5 km
- 9 km, 5 km
- 5 km, 9 km
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Answer: (c) 9 km, 5 km
From the figure: $AB = 5\,km$, $ED = BC = 3\,km$, $CD = 1\,km$, so $AE = 5 - 1 = 4\,km$.
Distance covered $= AB + BC + CD = 5 + 3 + 1 = 9\,km$
Displacement $= AD = \sqrt{AE^2 + ED^2} = \sqrt{4^2 + 3^2} = \sqrt{25} = 5\,km$
Important Points to Remember
- The magnitude of displacement is the smallest possible distance between two positions: distance ≥ displacement.
- Distance for a moving particle can never be zero. Displacement, however, can be zero, positive, or negative.
- Zero displacement means the particle has returned to its starting position after moving.
- Distance can never decrease over time for a moving object, but displacement can.
- When displacement decreases over time, the body is moving back toward its starting position.
