Distance and Displacement Concepts with solved Numerical and Work sheet PDF

 

Distance and Displacement Concepts with solved Numerical and Work sheet PDF

Distance:

• Definition: The total length of the actual path travelled by an object, regardless of direction.

• Nature/Type : Scalar quantity (only magnitude, no direction)

• Value : Always positive

• S.I. unit : meter (m)

• Formula : 

1. Distance = sum of length of path segments
2. Distance = Speed × time

Displacement :

• Definition : The shortest straight line distance between the initial position and final position of the object, along with direction.

• Nature/Type : Vector quantity (magnitude + direction)

• Value :  Can be positive, negative or zero

• S.I. unit : meter (m)

• Formula : 

1. Displacement = length of straight line between initial position and final position
2. Displacement = Final position - initial position
3. Displacement = Velocity × time
4. Displacement in 2D

If an object moves from

$$P_1(x_1, y_1) \quad \text{to} \quad P_2(x_2, y_2)$$

The displacement magnitude is:

$$\Delta s = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

The displacement vector is:

$$\overrightarrow{\mathrm{\Delta }s}=(x_2-x_1)\widehat{i}+(y_2-y_1)\widehat{j}$$

         5. Displacement in 3D

 If an object moves from

$$P_1(x_1, y_1, z_1) \quad \text{to} \quad P_2(x_2, y_2, z_2)$$

The displacement magnitude is:

$$\Delta s = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$$

The displacement vector is:

$$\vec{\Delta s} = (x_2 - x_1) \hat{i} + (y_2 - y_1) \hat{j} + (z_2 - z_1) \hat{k}$$

Key Tip:

• In 2D, use two components (x & y).

• In 3D, add the z-component to account for height or depth.

 Comparative study of Distance and displacement:  

Special cases:

1. straight line motion without change in direction:

                                Distance = Displacement

2. Return to starting point:    

 

                                Distance > 0 but displacement = 0

3. Curved path:

                            Distance > Displacement

keep remember: In any case distance will not less than the displacement

Difference between Distance and Displacement:

Features	Distance	Displacement
Definition	The total length of the actual path traveled by an object, regardless of direction.	The shortest straight line distance between the initial position and final position of the object, along with direction.
Type	Scalar quantity (only magnitude, no direction)	Vector quantity (magnitude + direction)
Value	Always positive	Can be positive, negative or zero
Depends on	Path taken	Initial and final positions only
Example: Walking 5 m east and 4 m west	Distance = 5 + 4 = 9 m	Displacement = 5 – 4 = 1 m

Solved Numerical :

Example 1: Straight Line Motion

A car travels 60 km east and then 40 km west.
Find:
(a) Total Distance
(b) Displacement

Solution:

• Distance = $60 + 40 = 100 \ \text{km}$

• Displacement = $60 - 40 = 20 \ \text{km east}$

Example 2: 2D Motion

A boy walks 3 km north, then 4 km east.
Find:
(a) Distance
(b) Displacement

Solution:

• Distance = $3 + 4 = 7 \ \text{km}$

• Displacement =

$$\sqrt{(4{)}^2+(3{)}^2}=\sqrt{16+9}=\sqrt{25}=5\text{ km}$$

Direction: $\tan\theta = \frac{4}{3} \implies \theta \approx 53.13^\circ$ east of north.

Example 3: Round Trip

A runner completes a 400 m circular track in 2 minutes.
Find:
(a) Distance
(b) Displacement

Solution:

• Distance = $400 \ \text{m}$

• Displacement = $0 \ \text{m}$ (start and end points are the same)

Example 4: Zig-zag motion

A footballer runs 20 m east, 10 m south, 20 m west and 10 m north.

Find:

(a) Distance
(b) Displacement

Solution:

• Distance = 20 + 10 + 20 + 10 = 60 m

• Displacement = 0 m (back to starting point)

Notes for Exams:

• Displacement magnitude ≤ Distance.
• Displacement = Distance only when path is a straight line without direction change.
• Always attach direction when giving displacement.
• For vector problems, use Pythagoras theorem for 2D and 3D distance formula for more dimensions.

Graph tips:

• In a distance-time graph, slope gives speed.
• In a displacement-time graph, slope gives velocity.

Multiple Choice Question on Distance and Displacement:

1) Distance is a:

                                         a) Scalar quantity

                                         b) Vector quantity

                                         c) Both scalar and vector

                                         d) None of these 

2) Displacement is a:

                                        a) Scalar quantity

                                        b) Vector quantity

                                        c) Constant quantity

                                        d) None of these

3) A boy walks 5 m east and then 5 m west. His displacement is:

                                        a) 10 m east

                                        b) 0 m

                                        c) 5 m

                                        d) 10 m west

4) A girl walks 3 m north, then 4 m east. The magnitude of her displacement is:

                                        a) 7 m

                                        b) 5 m

                                        c) 1 m

                                         d) 12 m

5) If an object returns to its starting point, then:

                                        a) Distance = 0, Displacement = 0

                                        b) Distance ≠ 0, Displacement = 0

                                        c) Distance = 0, Displacement ≠ 0

                                        d) Distance ≠ 0, Displacement ≠ 0

6)  The displacement of an object can be:

                                        a) Only positive

                                        b) Only negative

                                        c) Positive, negative, or zero

                                        d) Always zero

7) Distance between two points is always:

                                        a) Equal to displacement

                                        b) Greater than or equal to displacement

                                        c) Less than displacement

                                        d) None of these

8) A car travels 6 km north and then 8 km south. The displacement is:

                                        a) 14 km south

                                        b) 14 km north

                                        c) 2 km south

                                        d) 2 km north

9) A person walks around a circular park of circumference 400 m and comes back to the starting point. His displacement is:

                                        a) 400 m

                                        b) 0 m

                                        c) 200 m

                                        d) 800 m

10) Which of the following is always true?

                                        a) Distance ≥ Displacement

                                        b) Distance ≤ Displacement

                                        c) Distance = Displacement always

                                        d) Displacement > Distance

Answer Key

1)    a)	2)    b)	3)    b)	4)    b)	5)     b)
6)    c)	7)    b)	8)    c)	9)    b)	10) a)