Physics and Mathematics
Distance and Displacement
1. Introduction
In the study of motion, we often need to describe how far an object has moved and in which direction it has moved.
These two aspects are captured by two different quantities: Distance and Displacement.
👉 Understanding the difference between them is essential because they tell us different things about motion.
2. Distance
- Definition:
Distance is the total length of the actual path travelled by a moving object, regardless of the direction. - Key Features:
- Distance is always non-negative.
- It is a scalar quantity (depends only on magnitude, not direction).
- It cannot decrease with time.
- SI Unit: meter [m]
- Dimensional Formula: [ [L] ]
3. Displacement
- Definition:
Displacement is the shortest straight-line distance from the initial position to the final position of the object, along with the direction. - Key Features:
- Displacement is a vector quantity (has both magnitude and direction).
- It can be positive, negative, or zero depending on the direction of motion.
- Displacement is always ≤ distance for a given motion.
- SI Unit: metre [m]
- Dimensional Formula: [ [L] ]
4. Comparison Table
| Feature | Distance | Displacement |
|---|---|---|
| Type of quantity | Scalar | Vector |
| Depends on path | ✅ Yes | ❌ No (only initial & final points) |
| Can be negative? | ❌ No | ✅ Yes (if opposite to chosen direction) |
| Always increasing? | ✅ Yes | ❌ No (can decrease or become zero) |
| Value | ≥ 0 | ≤ distance |
5. Mathematical Relation
If an object moves from position [ x_i ] to [ x_f ], then:
- Displacement:
[ \Delta x = x_f – x_i ]
- Distance:
Total path length covered (depends on the route taken).
👉 For straight-line motion in one direction,
[ \text{Distance} = |\Delta x| ]
6. Examples
- A car moves 3 km east, then 4 km east again:
- Distance = 3 + 4 = 7 km
- Displacement = 7 km east
- A car moves 3 km east, then 4 km west:
- Distance = 3 + 4 = 7 km
- Displacement = 1 km east
- A runner starts at a point and comes back to the same point after running 400 m around a circular track:
- Distance = 400 m
- Displacement = 0 m
7. Practice Questions (With Solutions)
Conceptual Questions (10+)
1. Define distance.
Distance is the total length of the actual path travelled by an object.
It is a scalar quantity and is always non-negative. SI Unit: metre (m).
2. Define displacement.
Displacement is the shortest straight-line vector from the initial position
to the final position of the object. It has both magnitude and direction.
SI Unit: metre (m). If initial position is [x_i] and final is [x_f],
[Δx = x_f – x_i].
3. Is distance a scalar or vector quantity?
Distance is a scalar quantity — it has only magnitude and no direction.
4. Can displacement be greater than distance?
No. For any motion, the magnitude of displacement is always less than or equal
to the total distance travelled. [|Δx| ≤ distance].
5. Give one example where distance ≠ displacement.
A person walks 3 m east and then 4 m west. Distance = 7 m;
displacement = −1 m (1 m west).
6. Can distance be zero?
Only if there is no motion at all. Distance cannot be negative.
7. Can displacement be zero?
Yes. Displacement is zero when the final position equals the initial position
(e.g., a runner completes one lap and returns to the start).
8. If a particle moves in a straight line in one direction, how are distance and displacement related?
If motion is along a straight line without reversing direction, distance = |Δx|.
If motion is along the chosen positive axis, distance = Δx.
9. Can distance decrease with time during motion?
No. Distance is cumulative path length; it cannot decrease during motion.
10. Can displacement be negative? Explain.
Yes. Displacement is negative if the final position lies in the direction
opposite to the chosen positive axis.
11. A car moves 6 km north, then 8 km south. State its distance and displacement.
Distance = 6 + 8 = 14 km.
Displacement = 6 − 8 = −2 km (2 km south if north is positive).
12. Is it possible that distance remains the same but displacement changes?
For two different motions, yes — if start/end points differ but the total path
length is identical. For one fixed motion, distance and displacement are fixed.
Numerical Questions (With Solutions)
Q1: A boy walks 4 m east, then 3 m west. Find his distance and displacement.
- Solution:
[
\text{Distance} = 4 + 3 = 7 , \text{m}
]
[
\text{Displacement} = 4 – 3 = 1 , \text{m (east)}
]
Q2: A car goes 5 km east, then returns to the starting point. Find the distance and displacement.
- Solution:
[
\text{Distance} = 5 + 5 = 10 , \text{km}
]
[
\text{Displacement} = 0 , \text{km}
]
Q3: A person moves 6 m north, then 8 m east. Find the distance and displacement.
- Solution:
[
\text{Distance} = 6 + 8 = 14 , \text{m}
]
[
\text{Displacement} = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = 10 , \text{m (NE direction)}
]
8. FAQs / Common Misconceptions (10+)
Q1: Can distance ever be less than displacement?
❌ No. Distance is always ≥ displacement.
Q2: Can displacement be greater than distance?
❌ No, never.
Q3: Can displacement be zero even if distance is not zero?
✅ Yes, for example when an object returns to the starting point.
Q4: Why is distance called a scalar quantity?
✅ Because it has only magnitude.
Q5: Why is displacement a vector quantity?
✅ Because it has both magnitude and direction.
Q6: Can displacement be negative?
✅ Yes, if it is in the opposite direction of the chosen positive axis.
Q7: Does distance depend on the path taken?
✅ Yes.
Q8: Does displacement depend on the path taken?
❌ No, it only depends on the initial and final positions.
Q9: If an object moves in a circular path and comes back to the starting point, what is its displacement?
✅ Zero.
Q10: If distance is zero, what can we say about displacement?
✅ Displacement must also be zero — the object didn’t move.
9. Related Topics (Links)
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