Chapter 02 • Motion

Understanding Motion through Experience

Build a clear understanding of motion, including frame of reference, relative motion, scalars and vectors, graphical vector addition, equations of motion and distance travelled in the nth second.

01

Motion and Reference Frame

Understand motion, rest, reference points, inertial frames, non-inertial frames and relative motion.

02

Scalars and Vectors

Distinguish quantities having magnitude only from quantities having both magnitude and direction.

03

Vector Addition

Learn graphical addition of vectors using the triangle and parallelogram approaches.

04

Equations of Motion

Use the equations of uniformly accelerated motion and derive the distance travelled in the nth second.

Reflect Before You Begin

  • Why do we feel pushed backward when a bus suddenly starts moving?
  • Can an object be at rest for one observer but moving for another?
  • How do athletes decide the best angle to throw a ball so that it travels the maximum distance?
  • Can we measure motion using simple tools available in the classroom?

Discuss your ideas with classmates before beginning the activities.

Core Formulae

$$v=u+at$$
$$s=ut+\frac{1}{2}at^2$$
$$v^2=u^2+2as$$
$$s_n=u+\frac{a}{2}(2n-1)$$

Understanding Motion through Experience: Concepts

How to Learn This Chapter

Do not try to memorise the formulas first. First understand what is changing, what the observer is using as a reference, and whether direction matters. Then the formulas become much easier to use.

01

Identify the idea

Understand motion, reference frame, scalar, vector and resultant in simple language.

02

Connect the formula

Know what every symbol means and when the formula should be used.

03

See it visually

Use the interactive SVG demonstrations to connect equations with motion.

04

Check yourself

Attempt the Quick Check before opening the exercise solutions.

2.1 What Is Motion?

Definition

An object is in motion when its position changes with time relative to a chosen reference point.

Understand the idea

Motion is not decided by the object alone. We must first decide relative to what we are observing it. A person sitting inside a moving bus is at rest relative to another passenger, but moving relative to the road.

Key idea: Motion is a change of position with time relative to a reference point.

How do we describe motion?

The chapter connects motion with measurable quantities such as distance and time. Comparing the same journey over a known distance and time allows us to describe how quickly the motion occurs.

Simple Example

Two students cover the same 5 m distance. Student A takes 10 s and Student B takes 5 s. Student B completes the same distance in less time, so the motion is faster.

Quick Check — Motion

  1. What must change for an object to be called moving?
  2. Why is a reference point needed to describe motion?
  3. Can the same object be at rest for one observer and moving for another?
View Solutions in Exercise 2.1

2.2 Frame of Reference

Definition

A frame of reference is the reference point or reference system relative to which we describe the position and motion of an object.

Why is it necessary?

Saying “the car is moving” is incomplete. We must ask: moving relative to what? Relative to the road, the car may be moving. Relative to a passenger sitting inside it, another object may appear stationary.

Inertial Frame

A frame that is at rest or moving with constant velocity. Newton's laws hold without modification in such a frame.

Non-Inertial Frame

A frame that is accelerating. In such a frame, special corrections such as pseudo forces become necessary.

Remember: An accelerating observer and a non-accelerating observer can describe the same motion differently.

Think Like an Observer

Two passengers sit next to each other in a train moving at constant speed. Each passenger sees the other as at rest because their relative position does not change. A person standing beside the railway track sees both passengers moving.

3D Interactive Simulation: Frame of Reference
Three.js 3D Engine
Drag to orbit 3D view
Observer Frame: Roadside (Inertial)
Bus Speed: v = 4.0 m/s | Passenger appears moving relative to ground.
Switch Observer Frame:

Quick Check — Frame of Reference

  1. Define a frame of reference in your own words.
  2. What is an inertial frame?
  3. What is a non-inertial frame?
  4. Why can two observers describe the same object differently?
View Solutions in Exercise 2.1

2.3 Scalars and Vectors

Some physical quantities need only a numerical value and unit. Others also need a direction to describe them completely.

S

Scalar Quantity

Meaning: A physical quantity having magnitude only.

Examples: distance, time, mass, speed and work.

V

Vector Quantity

Meaning: A physical quantity having both magnitude and direction.

Examples: displacement, velocity and force.

Distance vs Displacement

Distance Displacement
Total path length travelled. Directed change from initial position to final position.
Scalar. Vector.
Depends on the actual path. Depends on initial and final positions.
Can be non-zero after returning to the starting point. Becomes zero when the final position is the same as the initial position.

Easy Way to Remember

Distance asks: “How much path did I cover?”
Displacement asks: “How far and in which direction am I from where I started?”

3D Interactive Simulation: Distance vs. Displacement
Three.js 3D Engine
Drag to orbit 3D track view
Track Progress: 0% | Actual Path Distance: 0.0 m | Direct Displacement: 0.0 m

Quick Check — Scalars and Vectors

  1. What is the defining difference between a scalar and a vector?
  2. Classify speed, velocity, distance, displacement, acceleration and mass.
  3. Give two everyday examples of vector quantities.
  4. What happens to displacement when an object returns to its starting point?
View Solutions in Exercise 2.1

2.4 Vector Addition — Graphical Method

Definition

Vector addition is the process of combining two or more vectors to obtain a single vector called the resultant.

Triangle Method

  1. Draw the first vector with the correct magnitude and direction.
  2. Place the tail of the second vector at the head of the first vector.
  3. Join the tail of the first vector to the head of the second vector.
  4. The joining vector is the resultant.
Direction matters. You cannot add vectors correctly by adding magnitudes alone unless their directions are also considered.
O (Start) Vector A (4 units East) Vector B (3 units North) Resultant R = √(4² + 3²) = 5 units
Figure 2.1: Triangle law of vector addition: Connecting tail of A to head of B gives resultant R = A + B.
3D Interactive Simulation: Triangle Law of Vector Addition
Three.js 3D Engine
Drag to orbit 3D vectors
Vector A = 4 units (East) | Vector B = 3 units (North)
Resultant: |R| = √(4² + 3²) = 5.00 units (θ = 36.9° North of East)
Vector A (East):
Vector B (North):

Worked Example — Sita's Journey

A(1,1), B(3,1), C(3,5) and D(4,5) are in km. Sita travels A → B on foot, then B → C → D by school bus.

(a) Distance on foot:

$$AB=\sqrt{(3-1)^2+(1-1)^2}=2\text{ km}$$

(b) Distance by bus:

$$BC=4\text{ km},\qquad CD=1\text{ km}$$ $$\text{Bus distance}=4+1=5\text{ km}$$

(c) Total displacement:

$$AD=\sqrt{(4-1)^2+(5-1)^2}=\sqrt{3^2+4^2}=5\text{ km}$$

Quick Check — Vector Addition

  1. What is a resultant vector?
  2. State the steps of the triangle method.
  3. For 4 units east and 3 units north, what is the resultant magnitude?
  4. What is the resultant of two equal vectors acting in opposite directions?
View Solutions in Exercise 2.1

2.5 Equations of Motion

When to use

These equations describe motion when an object moves with constant acceleration.

Symbols You Must Know

Symbol Meaning SI unit
$u$ Initial velocity m/s
$v$ Final velocity m/s
$a$ Acceleration m/s²
$t$ Time s
$s$ Displacement m

The Three Main Equations

Equation 1
$$\color{#0284c7}{v} = \color{#059669}{u} + \color{#e11d48}{a} \color{#7c3aed}{t}$$
v Final Velocity
u Initial Velocity
a Acceleration
t Time

Use this when the relationship between initial velocity, acceleration, time and final velocity is required.

Equation 2
$$\color{#0891b2}{s} = \color{#059669}{u}\color{#7c3aed}{t} + \frac{1}{2} \color{#e11d48}{a} \color{#7c3aed}{t}^2$$
s Displacement
u Initial Velocity
t Time
a Acceleration

Use this when displacement, initial velocity, acceleration and time are involved.

Equation 3
$$\color{#0284c7}{v}^2 = \color{#059669}{u}^2 + 2\color{#e11d48}{a}\color{#0891b2}{s}$$
v Final Velocity
u Initial Velocity
a Acceleration
s Displacement

Use this when time is not given or is not required.

Formula selection tip: If the question does not contain time, the third equation is often the most convenient. If time is given, the first or second equation may be more direct.

Understanding the Equations Instead of Memorising Them

The first equation tells us how velocity changes during constant acceleration. The second connects displacement with time. The third connects velocity and displacement without requiring time.

The supplied chapter gives these equations as the governing equations for constant acceleration; the detailed algebraic derivations are developed here as a learning aid.

Derivation of the First Equation

Acceleration is the change in velocity per unit time:

$$a=\frac{v-u}{t}$$

Multiply both sides by $t$:

$$at=v-u$$

Rearrange:

$$\boxed{v=u+at}$$

Derivation of the Second Equation

For constant acceleration, average velocity is the mean of initial and final velocities:

$$v_{\text{avg}}=\frac{u+v}{2}$$

Displacement equals average velocity × time:

$$s=\frac{u+v}{2}t$$

Using $v=u+at$:

$$s=\frac{u+(u+at)}{2}t$$ $$s=\frac{2u+at}{2}t$$ $$\boxed{s=ut+\frac12at^2}$$

Derivation of the Third Equation

Start with:

$$s=\frac{u+v}{2}t$$

From $v=u+at$:

$$t=\frac{v-u}{a}$$

Substitute:

$$s=\frac{u+v}{2}\cdot\frac{v-u}{a}$$ $$2as=(u+v)(v-u)=v^2-u^2$$ $$\boxed{v^2=u^2+2as}$$
3D Interactive Simulation: Uniform Acceleration ($v = u + at, s = ut + \frac{1}{2}at^2$)
Three.js 3D Engine
Drag to orbit 3D road view
Time: 0.0 s | Acceleration: a = 2.0 m/s² | Speed: v = 0.0 m/s | Displacement: s = 0.0 m
Acceleration (a):

Worked Example

A body starts from rest and accelerates at $4\,m/s^2$. Find the distance travelled in the 6th second.

Step 1 — Identify values:

$$u=0,\qquad a=4\,m/s^2,\qquad n=6$$

Step 2 — Use the nth-second formula:

$$s_n=u+\frac{a}{2}(2n-1)$$

Step 3 — Substitute:

$$s_6=0+\frac{4}{2}(12-1)=22\,m$$

Quick Check — Equations of Motion

  1. What condition must be satisfied before using these equations?
  2. What does $u$ represent? What does $v$ represent?
  3. Which equation is useful when time is not given?
  4. Write the first equation of motion.
View Solutions in Exercise 2.1

Distance Travelled in the nth Second

Important distinction

Distance travelled in the nth second means the distance covered during that one-second interval, not the total distance from the beginning.

Step 1 — Displacement in n seconds

$$s_n=un+\frac12an^2$$

Step 2 — Displacement in (n − 1) seconds

$$s_{n-1}=u(n-1)+\frac12a(n-1)^2$$

Step 3 — Subtract

The distance covered during the nth second is the difference between the displacement up to $n$ seconds and the displacement up to $(n-1)$ seconds:

$$s_{\text{nth second}}=s_n-s_{n-1}$$

Step 4 — Simplify

$$s_n-s_{n-1} = un+\frac12an^2-u(n-1)-\frac12a(n-1)^2$$
$$\boxed{\color{#0891b2}{s_{\text{nth second}}} = \color{#059669}{u} + \frac{\color{#e11d48}{a}}{2}(2\color{#7c3aed}{n}-1)}$$
s_nth Distance Covered in the nth Second
u Initial Velocity
a Uniform Acceleration
n Second Number (e.g. 5th, 6th)
Do not confuse: $s=ut+\frac12at^2$ gives displacement up to time $t$, whereas $s_{\text{nth second}}=u+\frac a2(2n-1)$ gives the displacement during the nth second.

Quick Example

If $u=8\,m/s$, $a=2\,m/s^2$ and $n=5$:

$$s_5=8+\frac22(2\times5-1)=8+9=\boxed{17\,m}$$

Quick Check — nth Second

  1. What is the difference between displacement in 5 s and distance travelled in the 5th second?
  2. Write the formula for distance travelled in the nth second.
  3. What does $n$ represent in this formula?
  4. Find the distance travelled in the 6th second when $u=0$ and $a=4\,m/s^2$.
View Solutions in Exercise 2.1

Chapter Connection — Put the Ideas Together

Motion

Position changes with time relative to a reference point.

Reference Frame

The observer's chosen reference determines how motion is described.

Scalar / Vector

Ask whether magnitude alone is enough or direction is also required.

Acceleration

When velocity changes uniformly, the equations of motion can describe the motion.

Exam strategy: First write the known quantities, then choose the correct formula, substitute with units, simplify carefully, and finally check whether the answer has the correct physical unit.

Exercise 2.1 – Understanding Motion through Experience

Question 1. Define a frame of reference in your own words.
View Answer

A frame of reference is the reference point or coordinate system relative to which the position and motion of an object are described.

Question 2. Give two real-life examples where motion depends on the observer.
View Answer

Examples include a passenger sitting in a moving train appearing at rest to another passenger but moving relative to the ground, and two people walking together at the same speed appearing at rest relative to each other.

Question 3. Why does a person sitting in a moving train appear at rest to another passenger?
View Answer

Both passengers have the same velocity relative to the train, so their relative position does not change.

Question 4. Classify the following as scalar or vector quantities: speed, velocity, displacement, distance, acceleration and mass.
View Answer

Scalars: speed, distance, mass.
Vectors: velocity, displacement, acceleration.

Question 5. Explain the difference between distance and displacement with an activity diagram.
View Answer

Distance is the total path length travelled and is a scalar quantity. Displacement is the directed change in position from the starting point to the final point and is a vector quantity.

In Activity 2.3, when a student walks from A to B and then returns to A, the distance is non-zero while the displacement is zero.

Question 6. Give two everyday examples of vector quantities.
View Answer

Displacement and velocity are two examples. Force is another vector quantity.

Question 7. Draw two vectors of 4 units east and 3 units north and find the resultant using the triangle method.
View Answer

Draw the 4-unit east vector first. From its head, draw the 3-unit north vector. Join the tail of the first vector to the head of the second. This joining vector is the resultant displacement.

For perpendicular vectors, its magnitude is $R=\sqrt{4^2+3^2}=5$ units.

Question 8. Explain how vector subtraction is performed graphically.
View Answer

Vector subtraction can be represented as addition of the negative of the vector being subtracted. Reverse the direction of the second vector and then add it graphically to the first vector.

Question 9. Draw two opposite vectors of equal magnitude. Calculate its resultant.
View Answer

Two equal vectors acting in exactly opposite directions cancel each other. Therefore, the resultant is zero.

Question 10. A body starts from rest and accelerates at 4 m/s². Find the distance travelled in the 6th second.
View Answer

Given $u=0$, $a=4\,m/s^2$, $n=6$.

$$s_n=u+\frac{a}{2}(2n-1)$$ $$s_6=0+\frac{4}{2}(12-1)=2\times11=22\,m$$
Question 11. A car with initial velocity 8 m/s accelerates at 2 m/s². Find the distance covered in the 5th second.
View Answer

Given $u=8\,m/s$, $a=2\,m/s^2$, $n=5$.

$$s_5=8+\frac{2}{2}(2\times5-1)$$ $$s_5=8+9=17\,m$$

Understanding Motion Worksheets – Class 9 Advanced Science

Worksheets

Worksheet content was not included in the supplied chapter text. This tab is reserved for printable worksheets on motion, reference frames, scalars and vectors, vector addition and equations of motion.

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Understanding Motion – Quick Revision Notes

Key Concepts

Concept Quick Revision
Motion An object is in motion if its position changes with time with respect to a reference point.
Frame of reference The reference point or system relative to which motion is described.
Inertial frame A frame at rest or moving with constant velocity in which Newton's laws hold without modification.
Non-inertial frame An accelerating frame in which special corrections such as pseudo forces become necessary.
Scalar A quantity having magnitude only.
Vector A quantity having both magnitude and direction.
Resultant The single vector obtained by combining two or more vectors.

Scalar and Vector Classification

Scalar Vector
Distance Displacement
Time Velocity
Mass Force
Speed Acceleration
Work

Equations of Motion

$$v=u+at$$
$$s=ut+\frac{1}{2}at^2$$
$$v^2=u^2+2as$$

Distance in the nth Second

$$s_n=u+\frac{a}{2}(2n-1)$$

Understanding Motion – Topic Tests

Interactive Tests

Test content was not included in the supplied chapter text. This tab is reserved for graded tests covering motion, reference frames, scalars and vectors, vector addition and equations of motion.

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