Physics Class 9 Chapter 1 Motion Full Chapter Book Lakhmir Singh and Manjit kaur with book Image

Chapter motion 

WHAT IS MOTION

 

Motion, in physics, change with time of the position or orientation of a body. Motion along a line or a curve is called translation. Motion that changes the orientation of a body is called rotation. In both cases all points in the body have the same velocity (directed speed) and the same acceleration (time rate of change of velocity). The most general kind of motion combines both translation and rotation.

हिंदी में

⇩

गति, भौतिकी में, शरीर की स्थिति या अभिविन्यास के समय के साथ बदलती है। किसी रेखा या वक्र के अनुदिश गति को अनुवाद कहते हैं। वह गति जो किसी पिंड के उन्मुखीकरण को बदल देती है, घूर्णन कहलाती है। दोनों ही मामलों में शरीर के सभी बिंदुओं का वेग (निर्देशित गति) और समान त्वरण (वेग के परिवर्तन की समय दर) समान होता है। सबसे सामान्य प्रकार की गति अनुवाद और रोटेशन दोनों को जोड़ती है।

 

 

Distance And Displacement

Distance and displacement are two quantities that seem to mean the same but are distinctly different with different meanings and definitions. Distance is the measure of “how much ground an object has covered during its motion” while displacement refers to the measure of “how far out of place is an object.” In this article, let us understand the difference between distance and displacement.

What is Distance?

Distance is the total movement of an object without any regard to direction. We can define distance as to how much ground an object has covered despite its starting or ending point.

Let’s understand the concept of distance with the help of the following diagram:

 

Explanation of distance

Distance here will be = 4m + 3m + 5m = 12 m

Distance Formula

Δd=d1+d2

How is Displacement defined?

Displacement is defined as the change in position of an object. It is a vector quantity and has a direction and magnitude. It is represented as an arrow that points from the starting position to the final position. For example- If an object moves from A position to B, then the object’s position changes. This change in position of an object is known as Displacement.

 

Displacement = Δx=xf−x0

xf

= Final Position

x0

= Initial Position

Δx

= Displacement

Examples of Distance and Displacement

Question 1. John travels 250 miles to North but then back-tracks to South for 105 miles to pick up a friend. What is John’s total displacement?

Answer: John’s starting position  Xi= 0.

Her final position Xf is the distance travelled N minus the distance South.

Calculating displacement, i.e.D.

D = ΔX = (Xf – Xi)

D = (250 mi N – 105 mi S) – 0

D = 145 mi N

Question 2. An object moves along the grid through points A, B, C, D, E, and F as shown below. The side of square tiles measures 0.5 km.

a) Calculate the distance covered by the moving object.

b) Find the magnitude of the displacement of the object.

 

Solution:

a) The distance covered by the moving object is calculated as follows:

AB + BC + CD + DE + EF

3 + 1 + 1.5 + 0.5 + 0.5 = 6.5 km

The distance covered by the moving object is 6.5 km.

b) The initial point is A and the final point is F, hence the magnitude of the displacement is equal to the distance AF which is calculated by applying Pythagoras’s theorem to the triangle AHF as shown in the figure below

 

Applying the Pythagorean formula, we get

AF2=AH2+HF2

Substituting the formula, we get

AF2=(0.5×4)2+(0.5×3)2=6.25 AF=√6.25km=2.5km

The magnitude of displacement is 2.5 km.

Distance vs Displacement

Distance	Displacement
The complete length of the path between any two points is called distance	Displacement is the direct length between any two points when measured along the minimum path between them
Distance is a scalar quantity as it only depends upon the magnitude and not the direction	Displacement is a vector quantity as it depends upon both magnitude and direction
Distance can only have positive values	Displacement can be positive, negative and even zero

 

The difference between scalar and vector quantities

Scalar and vector quantities

A quantity that has magnitude but no particular direction is described as scalar. A quantity that has magnitude and acts in a particular direction is described as vector.

Scalar quantities

Scalar quantities only have magnitude (size). 

For example, 11 m and 15 ms^-1 are both scalar quantities.

Scalar quantities include:

• distance ⇨ दूरी
• speed  ⇨ स्पीड
• time  ⇨  समय
• power ⇨  शक्ति
• energy  ⇨  ऊर्जा

Scalar quantities change when their magnitude changes.

Vector quantities

Vector quantities have both magnitude and direction. For example, 11 m east and 15 ms^-1 at 30° to the horizontal are both vector quantities.

 

Vector qualities include:

• displacement ⇨ विस्थापन
• velocity  ⇨ वेग
• acceleration  ⇨ त्वरण
• force  ⇨ बल
• weight  ⇨ वजन
• momentum  ⇨ गति

Vector quantities change when:

• their magnitude changes
• their direction changes
• their magnitude and direction both change

The difference between scalar and vector quantities is an important one.

Speed is a scalar quantity – it is the rate of change in the distance travelled by an object, while velocity is a vector quantity – it is the speed of an object in a particular direction.

Example

A geostationary satellite is in orbit above Earth. It moves at constant speed but its velocity is constantly changing (since its direction is always changing).

• the difference in two vectors quantities = final vector - initial vector
• the difference in two scalar quantities = large value - small value

Uniform Motion and Non-Uniform Motion

Uniform Motion:

 This type of motion is defined as the motion of an object in which the object travels in a straight line and its velocity remains constant along that line as it covers equal distances in equal intervals of time, irrespective of the duration of the time.

If a body is involved in rectilinear motion and the motion is consistent, then the acceleration of the body must be zero.

Example of Uniform Motion:

1. If the speed of a car is 10 m/s, it means that the car covers 10 meters in one second. The speed is constant in every second.
2. Movement of blades of a ceiling fan.

Non Uniform Motion:

This type of motion is defined as the motion of an object in which the object travels with varied speed and it does not cover same distance in equal time intervals, irrespective of the time interval duration.

Example of Non Uniform Motion:

1. If a car covers 10 meters in first two seconds, and 15 meters in next two seconds.
2. The motion of a train.

Now, people usually get confused between uniform motion and uniform acceleration. In the later phenomena, the object is having a constant acceleration in rectilinear motion, which means the object has different speed in every second, which clearly defines that motion is changing.

Difference  Between these two types of motions:

Comparison Parameters	Uniform Motion	Non Uniform Motion
Average Speed	The motion is similar to the actual speed of the object.	The motion is different from the actual speed of the object.
Rectilinear Motion	It has zero acceleration.	It has non-zero acceleration.
Graph	Distance-time graph shows a straight line	Distance-time graph shows a curved line
Distance	Covers equal distances in equal time interval.	Covers unequal distances in equal time interval.

What is Speed?

Speed

Speed is defined as

The rate of change of position of an object in any direction.

Speed is measured as the ratio of distance to the time in which the distance was covered. Speed is a scalar quantity as it has only direction and no magnitude.

Speed Formula

The formula of speed is given in the table below:

Speed = Distance/Time

s=dt

Where,

• s is the speed in m.s^-1
• d is the distance traveled in m
• t is the time taken in s

Speed Unit

Following are the units of speed are:

SI system	ms-1
Numerical of Book Images  Lakhmir singh and Manjit Kaur  (1) (2) (3) (4) (5)

What is Velocity?

The meaning of velocity of an object can be defined as the rate of change of the object’s position with respect to a frame of reference and time. It might sound complicated but velocity is basically speeding in a specific direction. It is a vector quantity, which means we need both magnitude (speed) and direction to define velocity. The SI unit of it is meter per second (ms^-1). If there is a change in magnitude or the direction in the velocity of a body the body is said to be accelerating.

Velocity Formula

The formula of velocity is given in the table below:

Velocity = Displacmant/Time

v=dt

Where,

• v is the Velocity in m.s^-1
• d is the displacement traveled in m
• t is the time taken in s

Initial and Final Velocity

Initial velocity describes how fast an object travels when gravity first applies force on the object. On the other hand, the final velocity is a vector quantity that measures the speed and direction of a moving body after it has reached its maximum acceleration.

How to find the final velocity?

Finding the final velocity is simple with a few calculations and basic conceptual knowledge.

1. Determine the object’s original velocity by dividing the time it took for the object to travel a given distance by the total distance. In the equation V = d/t, V is the velocity, d is the distance and t is the time.
2. Determine the object’s acceleration by dividing the object’s mass by the force and multiply the answer by the time it took for it to accelerate. For example, if the object weighs 30 kg and has a force of 15 N applied to it, then the acceleration would be 4 m/s.
3. Add the quantity obtained from Step 1 and Step 2 to obtain the final velocity. For example, if your initial velocity was 3 m/s and your object acceleration is 4 m/s, your final velocity is 7 m/s (3 + 4 = 7).

Constant Velocity

The motion with constant velocity is the simplest form of motion. We witness constant motion whenever an object slides over a horizontal, low friction surface (when a puck slides over a hockey rink.)

The above graph is a graph of displacement versus time for a body moving with constant velocity. The straight line in the graph can be algebraically represented as follows:
x=x0+vt In the equation, x₀ is the displacement at time t, v is the constant velocity of the body v=dxdt

.

Velocity Units

The SI unit of velocity is m/s (m⋅s−¹). Other units and dimensions of velocity are given in the table below.

Unit of velocity
Common symbols		v, v, →v
SI unit	m/s
Other units	mph, ft/s
Dimension	LT−1

Difference between Speed and Velocity

The detailed comparison in the tabular format is given below.

Velocity VS Speed
SPEED	VELOCITY
Speed is the quantitative measure of how quickly something is moving.	Velocity defines the direction of the movement of the body or the object.
Speed is primarily a scalar quantity	Velocity is essentially a vector quantity
It is the rate of change of distance	It is the rate of change of displacement
Speed of an object moving can never be negative	The velocity of a moving object can be zero.
Speed is a prime indicator of the rapidity of the object.	Velocity is the prime indicator of the position as well as the rapidity of the object.
It can be defined as the distance covered by an object in unit time.	Velocity can be defined as the displacement of the object in unit time.

What is Acceleration ?

Acceleration is defined as

The rate of change of velocity with respect to time.

Acceleration is a Vector quantity as it has both magnitude and direction. It is also the second derivative of position with respect to time or it is the first derivative of velocity with respect to time.

What is Acceleration Formula?

Acceleration formula is given as:

Acceleration = (finalvelocity)−(initialvelocity) time acceleration = changeinvelocitytime a=vf−vit

a=Δvt

Where,

• a is the acceleration in m.s^-2
• v_f is the final velocity in m.s^-1
• v_i is the initial velocity in m.s^-1
• t is the time interval in s
• Δv is the small change in the velocity in m.s^-1

What is the Unit of Acceleration?

The SI Unit of acceleration is given as:

SI unit

m/s2

 

Uniform and Non-uniform acceleration

So can we have a situation when speed remains constant but the body is accelerated? Actually, it is possible in circular where speed remains constant but since the direction is changing hence the velocity changes, and the body is said to be accelerated.

Average acceleration

The average acceleration over a period of time is defined as the total change in velocity in the given interval divided by the total time taken for the change. For a given interval of time, it is denoted as ā.

Where v₂ and v₁ are the instantaneous velocities at time t₂ and t₁ and ā is the average acceleration.

What is Instantaneous Acceleration?

Instantaneous acceleration is defined as

The ratio of change in velocity during a given time interval such that the time interval goes to zero.

What is the difference between Acceleration and Velocity?

Following is the table of acceleration vs velocity:

Parameter	Acceleration	Velocity
Definition	Acceleration is defined as the change in the velocity of an object with respect to time	Velocity is defined as the speed of an object in a particular direction
Formula	Velocity/Time	Displacement/Time
Unit	m.s-2	m.s-1

 

EQUATION OF UNIFORMLY ACCELERATED MOTION

(1) First Equation of Motion 

 (2) Second Equation of Motion 

(3) Third Equation of Motion

Motion and Rest

If the position of an object changes with respect to a reference point then it is said to be in motion wrt.that reference while if it does not changes then it is at rest wrt.that reference point. For the better understanding or to deal with the different situations of rest and motion we derive some standard equation relating terms distance,displacement,speed,velocity and acceleration of the body by the equation called as equations of motion.

Three Equations of Motion

In case of motion with uniform or constant acceleration (one with equal change in velocity in equal interval of time) we derive three standard equations of motion which are also known as the laws of constant acceleration. These equations contain quantities displacement(s), velocity (initial and final), time(t) and acceleration(a) that governs the motion of a particle. These equations can only be applied when acceleration of a body is constant and motion is a straight line. The three equations are,

• v = u + at
• v² = u² + 2as
• s = ut + ½at²

where, s = displacement; u = initial velocity; v = final velocity; a = acceleration; t = time of motion. 

 

Numerical 

Derivation of Equation of Motions

 

Now let's start the derivation with the first equation of motion i.e. v=u+at where u is the initial velocity, v is the final velocity and a is the constant acceleration.

Assuming that a body started with initial velocity “u” and after time t it acquires final velocity v due to uniform acceleration a. 

We know acceleration is defined as the rate of change of velocity, also which is given by slope of the velocity time graph.

Thus both from definition as well as graph  Acceleration = Change in velocity/Time Taken i.e. a = v-u /t or at = v-u

Therefore, we have: v = u + at

Now to derive the second equation again suppose a body is moving with initial velocity u after time t its velocity becomes v. The displacement covered by the during this interval of time is S and the acceleration of the body is represented by a.

Explanation: We know area under velocity time graph gives total displacement of the body thus area under velocity time graph is area of trapezium OABC.

Also area of trapezium = ½(sum of parallel sides)height

Sum of parallel sides=OA+BC=u+v and here,height=time interval t

Thus,area of trapezium = ½(u+v)t

Substituting v=u+at from first equation of motion we get,

Displacement =S =area of trapezium = ½(u+u+at)t

S = ½(2u+at)t=ut+½at²

Which is called the second equation of motion and is the relation between displacement S,initial velocity u,time interval t and acceleration a of the particle.

Now in order to derive the third equation again use 

Displacement =S =area of trapezium = ½(u+v)t

From first equation v=u+at we get v-u=at ⇒v-u/a=t

Substituting the value of t in S = ½(u+v)t 

We  get S=½(u+v)(v-u)/a=(v²-u²)/2a

⇒2as=v²-u²

⇒v² =u²+2as

Which is the third equation of motion and is the relation between final velocity v,initial velocity u,constant acceleration a and displacement S of the particle.

We can now also calculate the displacement of particles during the nth second, using these equations of motion derived above. In order to do so we will calculate the displacement covered in n seconds and subtract the displacement covered in n-1 seconds and to get the displacement in nth second

s_nth =S_n-s_n-1=un-u(n-1) + 1/2an²-1/2a(n-1)²

simplifying gives us final equation for displacement in the nth second is s = u + a(2n-1)/2

This equation is often regarded as modified form of second equation of motion

What is Uniform Circular Motion?

The movement of a body following a circular path is called a circular motion. Now, the motion of a body moving with constant speed along a circular path is called Uniform Circular Motion. Here, the speed is constant but the velocity changes.

 

If a particle is moving in a circle, it must have some acceleration acting towards the centre which is making it move around the centre. Since this acceleration is perpendicular to the velocity of a particle at every instant, it is only changing the direction of velocity and not magnitude and that’s why the motion is uniform circular motion. We call this acceleration centripetal acceleration (or radial acceleration), and the force acting towards the centre is called centripetal force.

In the case of uniform circular motion, the acceleration is:

ar = v2r = ω2r

If the mass of the particle is m, we can say from the second law of motion that:

F = ma

mv2r= mω2r

This is not a special force, actually force like tension or friction may be a cause of origination of centripetal force. When the vehicles turn on the roads, it is the frictional force between tyres and ground that provides the required centripetal force for turning.

NOTE

So if a particle is moving in a uniform circular motion:

1) Its speed is constant

2) Velocity is changing at every instant

3) There is no tangential acceleration

4) Radial (centripetal) acceleration = ω2r

5) v=ωr

In case of non-uniform circular motion, there is some tangential acceleration due to which the speed of the particle increases or decreases. The resultant acceleration is the vector sum of radial acceleration and tangential acceleration.

By Shreyansh singh