Describing Motion Around Us

Motion in a Straight Line

**Definition**: Motion in a straight line, also called linear motion, is the simplest form of motion where an object moves along a single straight path. Examples include a child swimming in a race, a vertically falling ball, a car on a highway, or a train on straight tracks.

Describing Position

To describe motion scientifically, we must first establish a **reference point** (origin) from which we measure the position of an object.

**Position** is described by specifying:

**Distance from reference point**: How far the object is from the origin

**Direction from reference point**: Which way it is located relative to origin

**Representing Direction on a Straight Line**:

Positions to the RIGHT of reference point O are marked as POSITIVE (+)

Positions to the LEFT of reference point O are marked as NEGATIVE (–)

**State of Motion**:

An object is **in motion** if its position with respect to the reference point changes with time

An object is **at rest** if its position with respect to the reference point does not change with time

**Important Distinction**:

An **instant of time** is a single reading on a clock at one point (e.g., t = 5 s)

A **time interval** is the duration between two instants of time (e.g., from t = 4 s to t = 10 s, the time interval is 6 s)

Distance Travelled and Displacement

Distance Travelled

**Definition**: Distance travelled is the total length of the path covered by an object, regardless of direction. It is a **scalar quantity** (only magnitude, no direction required).

**Characteristics**:

Always positive

Depends on the actual path followed

Cannot decrease with time (always remains equal or increases)

Uses SI unit: **metre (m)**

Displacement

**Definition**: Displacement is the **net change in position** of an object between two given instants of time. It is the straight-line distance between initial and final positions.

**Characteristics**:

**Vector quantity** (requires both magnitude AND direction)

Can be positive, negative, or zero

Requires direction specification using + or – sign

The magnitude of displacement is the shortest distance between initial and final positions

Uses SI unit: **metre (m)**

**Key Relationship**:

For motion WITHOUT turning back (in one direction only): **Distance = |Displacement|**

For motion WITH turning back: **Distance ≥ |Displacement|**

**Practical Example**:

Athlete starts at O (0 m), reaches point A (100 m) at t = 10 s, then runs back to point B (40 m) at t = 16 s.

Total distance travelled = OA + AB = 100 + 60 = **160 m**

Displacement from O to B = **40 m (in positive direction)**

These quantities are NOT equal because the athlete changed direction

**Activity Analysis**:

When a ball is thrown vertically upward from point O and returns to O:

Total distance = Up distance + Down distance

Displacement = **0 m** (returned to starting point)

Magnitude of displacement ≤ Total distance travelled (always)

Average Speed and Average Velocity

Average Speed

**Definition**: Average speed is the total distance travelled divided by the time interval during which the distance is covered.

**Formula**:

```

Average speed = Total distance travelled / Time interval

or

v_avg = d / t

```

**Characteristics**:

**Scalar quantity** (no direction, only magnitude)

Always positive

SI unit: **m s⁻¹** or **m/s** (also expressed as km h⁻¹)

Depends on the total path length, not just the endpoints

Uniform Motion vs Non-Uniform Motion

**Uniform Motion in Straight Line**: Object travels equal distances in equal time intervals (for ALL possible time intervals)

Constant speed throughout

Acceleration = 0

**Non-Uniform Motion in Straight Line**: Object travels unequal distances in equal time intervals

Speed increases, decreases, or both

Has acceleration

Average Velocity

**Definition**: Average velocity is the displacement (change in position) divided by the time interval in which the change occurs.

**Formulas**:

```

Average velocity (v_av) = Displacement / Time interval

v_av = s / t

```

Where:

s = displacement (change in position)

t = time interval

**Characteristics**:

**Vector quantity** (requires both magnitude AND direction)

Direction indicated by + or – sign (same as displacement direction)

Can be positive, negative, or zero

SI unit: **m s⁻¹** or **m/s**

Does NOT depend on path taken; depends only on initial and final positions

**Rate of Change Concept**:

Average velocity represents the **average rate of change of position** with respect to time.

**Important Relationship**:

Magnitude of average velocity = Average speed ONLY when object moves in one direction (no turning back)

When object reverses direction: Magnitude of average velocity < Average speed

**Example Problem**:

Sarang swims from one end of a 25 m pool to the other end and back in 50 seconds.

Total distance = 25 + 25 = 50 m

Displacement = 0 m (returned to starting point)

Average speed = 50 m / 50 s = **1 m s⁻¹**

Average velocity = 0 m / 50 s = **0 m s⁻¹**

Instantaneous Velocity

**Definition**: The velocity of an object at a particular instant of time (not over a time interval).

**Concept**: As the time interval around an instant becomes progressively smaller (approaching zero), the average velocity approaches a fixed value called instantaneous velocity. This concept becomes more important in higher grades when studying calculus-based physics.

**Practical Application**: The speedometer of a vehicle shows nearly the instantaneous speed at that moment, while the direction of the vehicle's motion (facing direction) shows the instantaneous velocity direction.

Average Acceleration

**Definition**: Average acceleration is the change in velocity divided by the time interval during which the change occurs.

**Formulas**:

```

Average acceleration = Change in velocity / Time interval

a = (v - u) / t

or

a = (Final velocity - Initial velocity) / Time interval

```

Where:

v = final velocity

u = initial velocity

t = time interval

**Characteristics**:

**Vector quantity** (requires both magnitude AND direction)

SI unit: **m s⁻²** (metre per second squared)

Direction indicated by + or – sign

Indicates HOW QUICKLY velocity is changing, NOT how fast object is moving

**Critical Point**: An object can be moving very fast yet have zero acceleration. Acceleration depends on the RATE OF CHANGE of velocity, not the velocity magnitude itself. Example: A bus traveling at constant 60 km/h on a straight highway has zero acceleration despite high speed.

Direction of Acceleration

**When magnitude of velocity INCREASES**:

Acceleration is in the SAME direction as velocity

Positive sign (+) when moving in positive direction and speeding up

Negative sign (–) when moving in negative direction and speeding up

**When magnitude of velocity DECREASES**:

Acceleration is OPPOSITE to the direction of velocity

Acts as a "retarding force" or "deceleration"

**Visual Representation**:

**Speeding up**: Acceleration vector and velocity vector point in same direction

**Slowing down**: Acceleration vector and velocity vector point in opposite directions

Uniform Acceleration

**Definition**: When magnitude of velocity changes by equal amounts in equal time intervals (for ALL possible time intervals), acceleration is constant/uniform.

**Example Problem**:

Bus moving at 36 km h⁻¹ (= 10 m s⁻¹) accelerates to 54 km h⁻¹ (= 15 m s⁻¹) in 10 seconds.

Calculation:

Initial velocity u = 10 m s⁻¹

Final velocity v = 15 m s⁻¹

Time t = 10 s

a = (v - u) / t = (15 - 10) / 10 = 0.5 m s⁻²

The acceleration is 0.5 m s⁻² in the direction of motion (since speed is increasing).

**Braking Example**:

Same bus moving at 54 km h⁻¹ (= 15 m s⁻¹) comes to stop in 5 seconds.

Calculation:

Initial velocity u = 15 m s⁻¹

Final velocity v = 0 m s⁻¹

Time t = 5 s

a = (0 - 15) / 5 = –3 m s⁻²

The negative sign indicates acceleration is opposite to velocity direction (retarding force).

Physical Feeling of Acceleration

When you sit in a vehicle:

**Sudden start from rest**: You feel a **jolt forward** (acceleration in direction of motion)

**Sudden braking**: You feel a **jolt backward** (acceleration opposite to motion direction)

**Constant velocity motion**: No jolt, zero acceleration

These sensations occur because your body experiences the effects of changing velocity.

Causes of Acceleration

Acceleration can result from:

**Change in magnitude of velocity** (speeding up or slowing down)

**Change in direction of velocity** (even if speed remains constant)

**Both changes simultaneously**

Later in the chapter (Section 4.4), we will study circular motion where acceleration occurs purely due to change in direction while speed remains constant.

India's Scientific Contributions to Motion Studies

Ancient and medieval Indian mathematicians understood the concept of speed and distance relationships centuries before modern physics formalized them.

**Historical Example from Ganitakaumudi (14th century CE)**:

Two postmen start walking toward each other from 210 yojanas apart. One travels 9 yojanas per day; the other covers 5 yojanas per day. When do they meet?

Solution:

Combined daily progress = 9 + 5 = 14 yojanas per day

Time to meet = 210 / 14 = 15 days

First postman covers 135 yojanas; second covers 75 yojanas

This demonstrates the ancient understanding that speed = distance / time, a fundamental concept still used today.

Practice Problems for Board Exams

**Problem Type 1**: Calculate average speed and velocity for round trips

**Problem Type 2**: Determine acceleration during acceleration and braking phases

**Problem Type 3**: Analyze motion graphs and interpret distance vs displacement

**Problem Type 4**: Apply kinematic equations to real-world scenarios involving vehicles

**Key Exam Points to Remember**:

Always distinguish between distance and displacement

Always include direction for velocity and acceleration answers

Unit conversion is essential: 1 m s⁻¹ = 3.6 km h⁻¹

Zero acceleration means constant velocity (not zero velocity)

Negative acceleration doesn't always mean slowing down; it means acceleration opposite to chosen positive direction