Newton's laws of motion full notes class 9 | cbse24

Table of Contents

1. Motion and Rest
2. Distance and Displacement
3. Uniform and Non Uniform motion
4. Speed Velocity and Acceleration
5. Equation of Uniformly accelerated motion
6. Graphical Representation of Motion
7. Speed-Time Graph when the Initial Speed of the Body is Not Zero
8. Derive the equation of motion by graphical Method
9. 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

Note:-1:-Displacement ≤ Distance

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

[3] Uniform and Non Uniform motion

A body has a uniform motion if it travels equal distances in equal intervals, no matter how small these time intervals may be. For example, a car running at a constant speed of say, 10 metres per second, will cover equal distances of 10 metres, every second, so its motion will be uniform.

A body has a non-uniform motion if it travels unequal distances in equal intervals of time. For example, if we drop a ball from the roof of a tall building, we will find that it covers unequal distances in equal intervals of time.

[4] Speed, Velocity and Acceleration

(1) Speed:-

The speed of a body is the distance travelled by it per unit time.

• 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 average speed of a body is the total distance travelled divided by the total time taken to cover this distance.

The formula for average speed is:

$$\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}$$

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:

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

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 $t$

3. Third Equation: $v^2 = u^2 + 2as$

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:

1. Displacement-Time Graph (s-t graph)
2. Velocity-Time Graph (v-t graph)
3. Acceleration-Time Graph (a-t graph)

1. Position-Time Graph:

• X-axis: Time
• Y-axis: Position (or distance)

Shape of graph:

.

2. Velocity-Time Graph:

• X-axis: Time
• Y-axis: Velocity

Shape of graph:

[7] Speed-Time Graph when the Initial Speed of the Body is Not Zero

To calculate acceleration from a speed-time graph, subtract the initial speed (OB) from the final speed (AC) and divide by time (OA). The distance travelled is the area under the graph, which forms a trapezium (OBCA). The distance is calculated as:

$$\text{Distance} = \frac{(OB + AC) \times OA}{2}$$

For non-uniform acceleration, the speed-time graph analysis becomes more complex.

[8] Derive the equation of motion by graphical Method

1. To Derive v = u + at by Graphical Method

Consider a velocity-time graph where a body starts with an initial velocity $u$ at point A, and its velocity increases uniformly to $v$ at point B over time $t$. The time is represented by OC, and the slope of line AB represents acceleration $a$.

From the graph:

• Initial velocity $u = OA$
  
• Final velocity $v = BC$
  

Since $BC = BD + DC$ and $DC = u$ , we have:

$$v = BD + u$$

The slope gives acceleration as:

$$a = \frac{BD}{t}$$

Thus, $BD = at$

Substitute this into the equation:

$$v=at+u\mathrm{<br>}\mathrm{<br>}$$

Rearranging, we get the first equation of motion:

$$v = u + at$$

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

To derive the second equation of motion, $s = ut + \frac{1}{2}at^2$  using a velocity-time graph 

Distance travelled is the area under the velocity-time graph (OABC), which consists of:

• Area of rectangle OADC:

  $$\text{Area}=u\times t=ut\mathrm{<br>}$$

• Area of triangle ABD:

$$\text{Area}=\frac{1}{2}\times t\times at=\frac{1}{2}a{t}^2$$

Total distance:
$$s=ut+\frac{1}{2}a{t}^2$$

This is the second equation of motion derived graphically

3:-derive the third equation of motion $v^2 = u^2 + 2as$

Distance travelled is the area under the velocity-time graph (OABC), which is a trapezium:

$$s=\frac{(u+v)\times t}{2}$$​

From the first equation of motion $v=u+at$ , rearranging gives:

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

Substituting $t$ into the equation for $s$:

$$s=\frac{(u+v)(v-u)}{2\mathrm{a}}$$

Simplifying gives:

$$2as=v^2-u^2$$

Thus, the third equation of motion is:

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

[9] Uniform Circular Motion

Uniform circular motion is the movement of an object along a circular path at a constant speed. While the speed remains unchanged, the object's velocity changes due to the continuous change in direction. This results in centripetal acceleration, directed toward the centre of the circle.

Key Points:

• Constant speed but changing velocity due to the changing direction.
• The centripetal force keeps the object in a circular path and is directed toward the centre.
• The magnitude of centripetal acceleration is

$$a_c=\frac{v^2}{r}$$ where $v$ is speed and $r$ is the radius.

Examples:

• A satellite orbiting Earth.
• A car on a circular track