## Table of Contents

**Motion and Rest****Distance and Displacement****Uniform and Non Uniform motion****Speed Velocity and Acceleration****Equation of Uniformly accelerated motion****Graphical Representation of Motion****Speed-Time Graph when the Initial Speed of the Body is Not Zero****Derive the equation of motion by graphical Method****Uniform Circular Motion**

## [1] Motion and Rest

**Motion:-**

A body is said to be in motion (or moving) if it changes its position relative to its surroundings over time.

**Example:-**Car moving w.r.t tree

**Rest:-**

A body is said to be at rest if it does not change its position with respect to its surroundings over time.

**Example:-**A book lying on a table

## [2] Distance and Displacement

**Distance:-** It has no specific direction

- Only magnitude
- SI unit- meter(m)
- CGS unit-(cm)
- Only positive
- Distance =AB+BC=5km+3km=8km
- Scalar

**Displacement:**-Shortes distance travelled

- Both magnitude and direction
- It may be positive, negative, or zero
- SI unit-meter(m)
- CGS unit-(cm)
- Displacement=AC=4km
- Vector

**Displacement ≤ DistanceNote:-1:-**

**2:-**Distance= Displacement (If body travels in a straight line)

## [3] Uniform and Non Uniform motion

**[4] Speed, Velocity and Acceleration**

**(1) Speed:-**

**Scalar****Positive**

The formula for speed is:

$$\text{Speed}=\frac{\text{Distance}}{\text{Time}}$$Where:

**Speed**is the rate at which an object moves.**Distance**is the total length of the path travelled.**Time**is the duration it takes to cover the distance.

**(a) Average Speed**:-

The formula for **average speed** is:

Where:

**Total Distance**is the sum of all distances travelled.**Total Time**is the total time taken to cover the entire distance.

**(b) Uniform Speed;-**

**Uniform speed** refers to the constant speed at which an object travels the same distance in equal intervals of time. In other words, the speed does not change over time.

For an object moving at uniform speed:

Since the speed is constant, the distance covered in each time interval remains the same. For example, if a car is moving at a uniform speed of 60 km/h, it will cover 60 kilometres every hour without any variation.

**(c) Non-uniform Speed:-**

**Non-uniform speed** refers to a condition where an object's speed changes over time. This means the object covers different distances in equal intervals of time.

**For example,** a car might cover 10 km in the first hour, 15 km in the second hour, and 5 km in the third hour. Since the distance covered per time interval is not constant, the speed is said to be non-uniform.

**(2) Velocity**

**Velocity** is a vector quantity that describes the rate at which an object changes its position, along with the direction of its movement. It is similar to speed but includes direction.

**Scalar****Positive, Negative or Zero**

The formula for velocity is:

$\text{Velocity} = \frac{\text{Displacement}}{\text{Time}}$Where:

**Displacement**is the straight-line distance between the starting and ending points, including direction.**Time**is the duration over which the displacement occurs.

Velocity is expressed in units such as meters per second (m/s) or kilometres per hour (km/h), and since it's a vector, it also includes a direction (e.g., 50 km/h north).

#### Difference between Speed and Velocity

### (3) Acceleration

**Acceleration** is the rate at which an object's velocity changes over time. It can refer to an increase or decrease in speed (sometimes called deceleration) or a change in direction.

The formula for acceleration is:

$\text{Acceleration} = \frac{\text{Change in Velocity}}{\text{Time Taken}}$

Or, more specifically:

$a = \frac{v_f - v_i}{t}$Where:

- $a$ = acceleration
- $v_f$ = final velocity
- $v_i$ = initial velocity
- $t$ = time taken for the change

### Units:

The SI unit of acceleration is **meters per second squared (m/s²)**

### Types:

**Positive acceleration**: Speed is increasing.**Negative acceleration or retardation (deceleration)**: Speed is decreasing.**Centripetal acceleration**: Change in direction, even if speed remains constant.

### Example:

If a car's velocity increases from 0 m/s to 20 m/s in 5 seconds, the acceleration is:

$a = \frac{20 \, \text{m/s} - 0 \, \text{m/s}}{5 \, \text{seconds}} = 4 \, \text{m/s}^2$

## [5] Equation of Uniformly accelerated motion

### 1. **First Equation: $v = u + at$**

**Definition of acceleration**:

$a = \frac{\text{Change in velocity}}{\text{Time taken}} = \frac{v - u}{t}$

Rearranging to solve for $v$:

$v = u + at$

This is the first equation of motion, which gives the final velocity after time $t$ under constant acceleration.

### 2. **Second Equation: $s = ut + \frac{1}{2}at^2$**

**Displacement** is the total distance covered. We can calculate it as the product of the average velocity and time. The average velocity for uniformly accelerated motion is:

$\text{Average velocity} = \frac{u + v}{2}$

Multiplying average velocity by time gives the displacement:

$s = \left(\frac{u + v}{2}\right) t$

Now, substitute $v$ from the first equation $v = u + at$

$s = \left(\frac{u + (u + at)}{2}\right) t$$s = \left(\frac{2u + at}{2}\right) t$$s = ut + \frac{1}{2}at^2$

This is the second equation of motion, which gives the displacement after time $t$

### $v^2 = u^2 + 2as$

**1. Start with the second equation of motion:**

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

**2. Use the first equation of motion v=u+at and solve for t:**

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

**3. Substitute t into the second equation:**

$s = u \left( \frac{v - u}{a} \right) + \frac{1}{2}a \left( \frac{v - u}{a} \right)^2$

**4. Simplify the expression:**

$s = \frac{v^2 - u^2}{2a}$

**5. Multiply by 2a to get:**

$v^2 = u^2 + 2as$

**Note:- (1)** If a body starts from rest.its initial velocity** u=0**

**(2)** If a body comes to rest(stops) , its final velocity **v=0**

**(3)** If a body moves with uniform velocity, its acceleration,**a=0**

## [6]Graphical Representation of Motion

The motion of an object can be represented graphically using various types of graphs. The most common ones include:

**Displacement-Time Graph**(s-t graph)**Velocity-Time Graph**(v-t graph)**Acceleration-Time Graph**(a-t graph)

2:-To derive the second equation of motion, $s = ut + \frac{1}{2}at^2$

2:-To derive the second equation of motion, $s = ut + \frac{1}{2}at^2$