SCALARS AND VECTORS

 Introduction to vectors and scalars 

We come into contact with many physical quantities in the natural world on a daily basis. For example, things like time, mass, weight, force, and electric charge, are physical quantities with which we are all familiar. We know that time passes and physical objects have mass. Things have weight due to gravity. We exert forces when we open doors, walk along the street and kick balls. We experience electric charge directly through static shocks in winter and through using anything which runs on electricity.

There are many physical quantities in nature, and we can divide them up into two broad groups called vectors and scalars.

Scalars and vectors

Scalars are physical quantities which have only a number value or a size (magnitude). A scalar tells you how much of something there is. 

A scalar is a physical quantity that has only a magnitude (size).

For example, a person buys a tub of margarine which is labelled with a mass of 500 g. The mass of the tub of margarine is a scalar quantity. It only needs one number to describe it, in this case,
500 g.

Vectors are different because they are physical quantities which have a size and a direction. A vector tells you how much of something there is and which direction it is in. Vector

A vector is a physical quantity that has both a magnitude and a direction.

For example, a car is travelling east along a freeway at
100 km⋅h−1. What we have here is a vector called the velocity. The car is moving at 100 km⋅h−1 (this is the magnitude) and we know where it is going – east (this is the direction). These two quantities, the speed and direction of the car, (a magnitude and a direction) together form a vector we call velocity.

Examples of scalar quantities:

1. mass has only a value, no direction

2. electric charge has only a value, no direction


Examples of vector quantities:

1. force has a value and a direction. You push or pull something with some strength (magnitude) in a particular direction

2. weight has a value and a direction. Your weight is proportional to your mass (magnitude) and is always in the direction towards the centre of the earth.

Vector notation

Vectors are different to scalars and must have their own notation. There are many ways of writing the symbol for a vector. In this book vectors will be shown by symbols with an arrow pointing to the right above it. For example,

⃗ F→, ⃗ W→ and ⃗ v→ represent the vectors of force, weight and velocity, meaning they have both a magnitude and a direction.

Sometimes just the magnitude of a vector is needed. In this case, the arrow is omitted. For the case of the force vector:

⃗ F→ represents the force vector

F represents the magnitude of the force vector

Graphical representation of vectors

Vectors are drawn as arrows. An arrow has both a magnitude (how long it is) and a direction (the direction in which it points). The starting point of a vector is known as the tail and the end point is known as the head.

 Examples of vectors

Parts of a vector

Directions 

There are many acceptable methods of writing vectors. As long as the vector has a magnitude and a direction, it is most likely acceptable. These different methods come from the different methods of representing a direction for a vector.
Relative directions 

The simplest way to show direction is with relative directions: to the left, to the right, forward, backward, up and down. 

Compass directions 

Another common method of expressing directions is to use the points of a compass: North, South, East, and West. If a vector does not point exactly in one of the compass directions, then we use an angle. For example, we can have a vector pointing 40°° North of West. Start with the vector pointing along the West direction (look at the dashed arrow below), then rotate the vector towards the north until there is a 40° angle between the vector and the West direction (the solid arrow below). The direction of this vector can also be described as: W 40°N (West 40°° North); or N 50° W (North 50° West).

Bearing

A further method of expressing direction is to use a bearing. A bearing is a direction relative to a fixed point. Given just an angle, the convention is to define the angle clockwise with respect to North. So, a vector with a direction of 110°° has been rotated clockwise 110°° relative to North. A bearing is always written as a three digit number, for example 275° or 080080° (for 8080°).

Drawing vectors

In order to draw a vector accurately we must represent its magnitude properly and include a reference direction in the diagram. A scale allows us to translate the length of the arrow into the vector's magnitude. For instance if one chooses a scale of 1 cm = 2 N(1 cm represents 2 N), a force of 20 N towards the East would be represented as an arrow 10 cm long pointing towards the right. The points of a compass are often used to show direction or alternatively an arrow pointing in the reference direction.

Method: Drawing Vectors

1. Decide upon a scale and write it down.

2. Decide on a reference direction

3. Determine the length of the arrow representing the vector, by using the scale.

4. Draw the vector as an arrow. Make sure that you fill in the arrow head.

5. Fill in the magnitude of the vector. 

WORKED EXAMPLE 1: DRAWING VECTORS 1

1. Draw the following vector quantity: v⃗ =6 m⋅s−1v→=6 m·s−1 North

2. Decide on a scale and write it down. 1 cm = 2 m⋅s−1

3. Decide on a reference direction


4. Determine the length of the arrow at the specific scale.

If  1 cm = 2 m⋅s−1, then  6 m⋅s−1 = 3 cm

5. Draw the vector as an arrow.
Scale used: 1 cm = 2 m⋅s−1

WORKED EXAMPLE 2: DRAWING VECTORS 2

Draw the following vector quantity: s⃗ =16 ms→=16 m east

Decide on a scale and write it down.

11 cm = 44 m

Decide on a reference direction

Determine the length of the arrow at the specific scale.

If  1 cm = 4 m, then  16 m = 4 cm

Draw the vector as an arrow

Scale used: 1 cm = 4 m

Direction = East

  

Properties of vectors

Vectors are mathematical objects and we will now study some of their mathematical properties.

If two vectors have the same magnitude (size) and the same direction, then we call them equal to each other. For example, if we have two forces, F1→=20 N in the upward direction and F2→=20 N in the upward direction, then we can say that F1→=F2→. Equality of vectors 

Two vectors are equal if they have the same magnitude and the same direction.

Just like scalars which can have positive or negative values, vectors can also be positive or negative. A negative vector is a vector which points in the direction opposite to the reference positive direction. For example, if in a particular situation, we define the upward direction as the reference positive direction, then a force F1→=30 N downwards would be a negative vector and could also be written as F1→=−30 N. In this case, the negative sign (−) indicates that the direction of F1→ is opposite to that of the reference positive direction. Negative vector 

A negative vector is a vector that has the opposite direction to the reference positive direction.

Like scalars, vectors can also be added and subtracted. We will investigate how to do this next.

Addition and subtraction of vectors

Adding vectors

When vectors are added, we need to take into account both their magnitudes and directions. 

For example, imagine the following. You and a friend are trying to move a heavy box. You stand behind it and push forwards with a force F1→ and your friend stands in front and pulls it towards them with a force F2→. The two forces are in the same direction (i.e. forwards) and so the total force acting on the box is:

It is very easy to understand the concept of vector addition through an activity using the displacement vector.
Displacement is the vector which describes the change in an object's position. It is a vector that points from the initial position to the final position. 

Adding vectors 

Materials

1.masking tape
2.Method

Tape a line of masking tape horizontally across the floor. This will be your starting point.

Task 1:

Take 2 steps in the forward direction. Use a piece of masking tape to mark your end point and label it A. Then take another 3 steps in the forward direction. Use masking tape to mark your final position as B. Make sure you try to keep your steps all the same length!

Task 2:

Go back to your starting line. Now take 3 steps forward. Use a piece of masking tape to mark your end point and label it B. Then take another 2 steps forward and use a new piece of masking tape to mark your final position as A.

Discussion

What do you notice?

In Task 1, the first 2 steps forward represent a displacement vector and the second 3 steps forward also form a displacement vector. If we did not stop after the first 2 steps, we would have taken 5 steps in the forward direction in total. Therefore, if we add the displacement vectors for 2 steps and 3 steps, we should get a total of 5 steps in the forward direction.

It does not matter whether you take 3 steps forward and then 2 steps forward, or two steps followed by another 3steps forward. Your final position is the same! The order of the addition does not matter!

We can represent vector addition graphically, based on the activity above. Draw the vector for the first two steps forward, followed by the vector with the next three steps forward.

We add the second vector at the end of the first vector, since this is where we now are after the first vector has acted. The vector from the tail of the first vector (the starting point) to the head of the second vector (the end point) is then the sum of the vectors.
As you can convince yourself, the order in which you add vectors does not matter. In the example above, if you decided to first go 3steps forward and then another 2 steps forward, the end result would still be 5 steps forward.

Subtracting vectors

Let's go back to the problem of the heavy box that you and your friend are trying to move. If you didn't communicate properly first, you both might think that you should pull in your own directions! Imagine you stand behind the box and pull it towards you with a force F1→ and your friend stands at the front of the box and pulls it towards them with a force F2→. In this case the two forces are in opposite directions. If we define the direction your friend is pulling in as positive then the force you are exerting must be negative since it is in the opposite direction. We can write the total force exerted on the box as the sum of the individual forces:

What you have done here is actually to subtract two vectors! This is the same as adding two vectors which have opposite directions.

As we did before, we can illustrate vector subtraction nicely using displacement vectors. If you take 5 steps forward and then subtract 3 steps forward you are left with only two steps forward:

What did you physically do to subtract 3 steps? You originally took 5 steps forward but then you took 3 steps backward to land up back with only 2 steps forward. That backward displacement is represented by an arrow pointing to the left (backwards) with length 3. The net result of adding these two vectors is 2 steps forward:

Thus, subtracting a vector from another is the same as adding a vector in the opposite direction (i.e. subtracting 3 steps forwards is the same as adding 3 steps backwards). 

Subtracting a vector from another is the same as adding a vector in the opposite direction. 

The resultant vector 

The final quantity you get when adding or subtracting vectors is called the resultant vector. In other words, the individual vectors can be replaced by the resultant – the overall effect is the same. Resultant vector 

The resultant vector is the single vector whose effect is the same as the individual vectors acting together. 

We can illustrate the concept of the resultant vector by considering our two situations in using forces to move the heavy box. In the first case (on the left), you and your friend are applying forces in the same direction. The resultant force will be the sum of your two applied forces in that direction. In the second case (on the right), the forces are applied in opposite directions. The resultant vector will again be the sum of your two applied forces, however after choosing a positive direction, one force will be positive and the other will be negative and the sign of the resultant force will just depend on which direction you chose as positive. For clarity look at the diagrams below. 

Forces are applied in the same direction (positive direction to the right)

Forces are applied in opposite directions (positive direction to the right)

There is a special name for the vector which has the same magnitude as the resultant vector but the opposite direction: the equilibrant. If you add the resultant vector and the equilibrant vectors together, the answer is always zero because the equilibrant cancels the resultant out. 

Equilibrant 

The equilibrant is the vector which has the same magnitude but opposite direction to the resultant vector. 

If you refer to the pictures of the heavy box before, the equilibrant forces for the two situations would look like:

 Techniques of vector addition 

Now that you have learned about the mathematical properties of vectors, we return to vector addition in more detail. There are a number of techniques of vector addition. These techniques fall into two main categories - graphical and algebraic techniques.

Graphical techniques 

Graphical techniques involve drawing accurate scale diagrams to denote individual vectors and their resultants. We will look at just one graphical method: the head-to-tail method.

Method: Head-to-Tail Method of Vector Addition

1. Draw a rough sketch of the situation. 

2. Choose a scale and include a reference direction. 

3. Choose any of the vectors and draw it as an arrow in the correct direction and of the correct length – remember to put an arrowhead on the end to denote its direction. 

4. Take the next vector and draw it as an arrow starting from the arrowhead of the first vector in the correct direction and of the correct length. 

5. Continue until you have drawn each vector – each time starting from the head of the previous vector. In this way, the vectors to be added are drawn one after the other head-to-tail. 

6. The resultant is then the vector drawn from the tail of the first vector to the head of the last. Its magnitude can be determined from the length of its arrow using the scale. Its direction too can be determined from the scale diagram. 

Let's consider some more examples of vector addition using displacements. The arrows tell you how far to move and in what direction. Arrows to the right correspond to steps forward, while arrows to the left correspond to steps backward. Look at all of the examples below and check them. 

This example says 1 step forward and then another step forward is the same as an arrow twice as long – two steps forward.

This example says 1 step backward and then another step backward is the same as an arrow twice as long – two steps backward.

It is sometimes possible that you end up back where you started. In this case the net result of what you have done is that you have gone nowhere (your start and end points are at the same place). In this case, your resultant displacement is a vector with length zero units. We use the symbol 

0→

Check the following examples in the same way. Arrows up the page can be seen as steps left and arrows down the page as steps right. 

Try a couple to convince yourself!

It is important to realise that the directions are not special– forward and backwards'or forward and backwards′or left and right' are treated in the same way. The same is true of any set of parallel directions:

In the above examples the separate displacements were parallel to one another. However the same head-to-tail technique of vector addition can be applied to vectors in any direction. 

WORKED EXAMPLE 3: HEAD-TO-TAIL ADDITION 1

A car breaks down in the road and you and your friend, who happen to be walking past, help the driver push-start it. You and your friend stand together at the rear of the car. If you push with a force of 50 N and your friend pushes with a force of
45 N, what is the resultant force on the car? Use the head-to-tail technique to calculate the answer graphically.

1. Draw a rough sketch of the situation



2. Choose a scale and a reference direction

Let's choose the direction to the right as the positive direction. The scale can be
1 cm =
10 N.

3. Choose one of the vectors and draw it as an arrow of the correct length in the correct direction


Start with your force vector and draw an arrow pointing to the right which is
5 cm long (i.e.
50 N = 5 ×10 N, therefore, you must multiply your cm scale by
5 as well).

4. Take the next vector and draw it starting at the arrowhead of the previous vector.

Since your friend is pushing in the same direction as you, your force vectors must point in the same direction. Using the scale, this arrow should be
4,5 cm long.



5. Draw the resultant, measure its length and find its direction

There are only two vectors in this problem, so the resultant vector must be drawn from the tail (i.e. starting point) of the first vector to the head of the second vector.



The resultant vector measures 95 cm and points to the right. Therefore the resultant force must be 95 N in the positive direction (or to the right). 

WORKED EXAMPLE 4: HEAD-TO-TAIL ADDITION 2

Use the graphical head-to-tail method to determine the resultant force on a rugby player if two players on his team are pushing him forwards with forces of F1→=60 N and F2→=90 N respectively and two players from the opposing team are pushing him backwards with forces of F3→=100 N and F4→=65 N respectively.

1. Choose a scale and a reference direction


Let's choose a scale of 0,5cm= 10N and for our diagram we will define the positive direction as to the right.

2. Choose one of the vectors and draw it as an arrow of the correct length in the correct direction

We will start with drawing the vector F1→=60 N, pointing in the positive direction. Using our scale of 0,5 cm = 10 N, the length of the arrow must be 3 cm pointing to the right.



3. Take the next vector and draw it starting at the arrowhead of the previous vector

The next vector is F2→=90 N in the same direction as F1→. Using the scale, the arrow should be
4,5 cm long and pointing to the right.



Take the next vector and draw it starting at the arrowhead of the previous vector

The next vector is F3→=100 N in the opposite direction. Using the scale, this arrow should be 5 cm long and point to the left.

Note: We are working in one dimension so this arrow would be drawn on top of the first vectors to the left. This will get messy so we'll draw it next to the actual line as well to show you what it looks like.



Take the next vector and draw it starting at the arrowhead of the previous vector

The fourth vector is F4→=65 N also in the opposite direction. Using the scale, this arrow must be
3,25 cm long and point to the left.



Draw the resultant, measure its length and find its direction


We have now drawn all the force vectors that are being applied to the player. The resultant vector is the arrow which starts at the tail of the first vector and ends at the head of the last drawn vector.




The resultant vector measures 0,75 cm which, using our scale is equivalent to 15 N and points to the left (or the negative direction or the direction the opposing team members are pushing in). Algebraic techniques Vectors in a straight line

Whenever you are faced with adding vectors acting in a straight line (i.e. some directed left and some right, or some acting up and others down) you can use a very simple algebraic technique:

Method: Addition/Subtraction of Vectors in a Straight Line


1. Choose a positive direction. As an example, for situations involving displacements in the directions west and east, you might choose west as your positive direction. In that case, displacements east are negative.

2. Next simply add (or subtract) the magnitude of the vectors using the appropriate signs.

3. As a final step the direction of the resultant should be included in words (positive answers are in the positive direction, while negative resultants are in the negative direction).


Let us consider a few examples. 

Remember that the technique of addition and subtraction just discussed can only be applied to vectors acting along a straight line. When vectors are not in a straight line, i.e. at an angle to each other then simple geometric and trigonometric techniques can be used to find resultant vectors.

VISCOSITY

Viscosity refers to resistance of a fluid (liquid or gas) to a change in shape, or movement of neighboring portions relative to one another. Viscosity denotes opposition to flow. The reciprocal of the viscosity is called the fluidity, a measure of the ease of flow. Molasses, for example, has a greater viscosity than water. Because part of a fluid that is forced to move carries along to some extent adjacent parts, viscosity may be thought of as internal friction between the molecules; such friction opposes the development of velocity differences within a fluid. Viscosity is a major factor in determining the forces that must be overcome when fluids are used in lubrication and transported in pipelines. It controls the liquid flow in such processes as spraying, injection molding, and surface coating.

A simulation of liquids with different viscosities. The liquid on the right has higher viscosity than the liquid on the left.

For many fluids the tangential, or shearing, stress that causes flow is directly proportional to the rate of shear strain, or rate of deformation, that results. In other words, the shear stress divided by the rate of shear strain is constant for a given fluid at a fixed temperature. This constant is called the dynamic, or absolute, viscosity and often simply the viscosity. Fluids that behave in this way are called Newtonian fluids in honour of Sir Isaac Newton, who first formulated this mathematical description of viscosity.

The dimensions of dynamic viscosity are force × time ÷ area. The unit of viscosity, accordingly, is newton-second per square metre, which is usually expressed as pascal-second in SI units.

The viscosity of liquids decreases rapidly with an increase in temperature, and the viscosity of gases increases with an increase in temperature. Thus, upon heating, liquids flow more easily, whereas gases flow more sluggishly. For example, the viscosities of water at 27 °C (81 °F) and at 77 °C (171 °F) are 0.85 × 10−3 and 0.36 × 10−3 pascal-second, respectively, but those of air at the same temperatures are 1.85 × 10−5 and 2.08 × 10−5 pascal-second.

For some applications the kinematic viscosity is more useful than the absolute, or dynamic, viscosity. Kinematic viscosity is the absolute viscosity of a fluid divided by its mass density. (Mass density is the mass of a substance divided by its volume.) The dimensions of kinematic viscosity are area divided by time; the appropriate units are metre squared per second. The unit of kinematic viscosity in the centimetre-gram-second (CGS) system, called the stokes in Britain and the stoke in the U.S., is named for the British physicist Sir George Gabriel Stokes. The stoke is defined as one centimetre squared per second.

FRICTION

WHAT IS FRICTION?

Friction is a catchall word that refers to any force that resists relative tangential motion (or intended motion). "Relative tangential motion" is a fancy way to say "slipping" or "sliding". Its direction is opposite the relative velocity (or intended velocity). Therefore, friction is defined as a force which opposes the relative motion between two or more surfaces that are contact.

TYPES OF FRICTION?

There are four types of friction: static, sliding, rolling, and fluid friction. Static, sliding, and rolling friction occur between solid surfaces. Static friction is strongest, followed by sliding friction, and then rolling friction, which is weakest. Fluid friction occurs in fluids, which are liquids or gases.

NOTE: SLIDING FRICTION, ROLLING FRICTION AND FLUID FRICTION ARE TYPES OF DYNAMIC FRICTION. 

STATIC FRICTION

Static friction exists between a stationary object and the surface on which it is resting. It prevents an object from moving against the surface.

Examples: static friction prevents an object like book falling from the desk even if the desk is slightly tilted, it helps us to pick up an object without slipping through our fingers.

When we want to move an object first we must overcome the static friction acting between the object and the surface on which the object is resting.

A stationary book on surface

SLIDING FRICTION

Sliding friction occurs between objects as the slide against each other. It acts in the direction opposite to the direction of motion. It prevents the object from moving too fast. It is also called as kinetic friction.

Examples: A book sliding from an inclined desk, A kid sliding on a slide.

When sliding friction is acting there must be another force existing to keep the body in moving, in case of the book sliding from the desk the other force acting is gravitational force.

children sliding on a slide

ROLLING FRICTION

Rolling friction hinders the motion of an object rolling on a surface, that means it slows down the motion of an object rolling on a surface.

Examples: It slows down a ball rolling on a surface and it slows down the motion of tire rolling on the surface.

Like sliding friction here also another force is required to keep the object in motion, in case of pedaling bicycle the bicyclist provides the force which is required for the bicycle to be in motion.

A ball rolling on a surface

FLUID FRICTION

Fluid friction is experienced by the objects moving through a fluid.Fluid friction acts between the object and the fluid through which it is moving.It is also called as drag.This force depends upon the object's shape, material, speed with which it is moving and the viscosity of the fluid. Viscosity is the measure of resistance of the fluid to flow and it differs from one fluid to other.

Examples: It slow downs the motion of airplane flying in the air, here the airplanes engine helps the plane to overcome the fluid friction and move forward.

A plane in air

NOTE: SLIDING FRICTION, ROLLING FRICTION AND FLUID FRICTION ARE TYPES OF DYNAMIC FRICTION. 

LAWS OF FRICTION

THE FIVE LAWS OF FRICTION 

1. When an object is moving, the friction is proportional and perpendicular to the normal force (N)

2. Friction is independent of the area of contact so long as there is an area of contact.

3. The coefficient of static friction is slightly greater than the coefficient of kinetic friction.

4. Within rather large limits, kinetic friction is independent of velocity.

5. Friction depends upon the nature of the surfaces in contact.

6. Friction is the force which opposes the relative motion between two or more surfaces that are contact

7. The ratio of the limiting frictional force is and the normal reaction equals a constant which is known as the co-efficient of static friction

ADVANTAGES OF FRICTION       

Friction plays a vital role in our daily life. Without friction we are handicap.

1. It is becomes difficult to walk on a slippery road due to low friction. When we move on ice, it becomes difficult to walk due to low friction of ice.

2. We cannot fix nail in the wood or wall if there is no friction. It is friction which holds the nail.

3. A horse cannot pull a cart unless friction furnishes him a secure Foothold.

DISADVANTAGES OF FRICTION  

Despite the fact that the friction is very important in our daily life, it also has some disadvantages like:

1. The main disadvantage of friction is that it produces heat in various parts of machines. In this way some useful energy is wasted as heat energy.

2. Due to friction we have to exert more power in machines.

3. It opposes the motion.

4. Due to friction, noise is also produced in machines.

5. Due to friction, engines of automobiles consume more fuel which is a money loss.

METHODS OF REDUCING FRICTION

There are a number of methods to reduce friction in which some are discussed here.

1. USE OF LUBRICANTS:

The parts of machines which are moving over one another must be properly lubricated by using oils and lubricants of suitable viscosity.

2. USE OF GREASE:

Proper greasing between the sliding parts of machine reduces the friction.

3. USE OF BALL BEARING:

In machines where possible, sliding friction can be replaced by rolling friction by using ball bearings.

4. DESIGN MODIFICATION:

Friction can be reduced by changing the design of fast moving objects. The front of vehicles and airplanes made oblong to minimize friction.

    

  

MOTION

When we observe our surroundings, we can see many physical interactions taking place around us like a book falling, an ear drum vibrating, bus moving, nuclear reactions etc. Everything in the universe moves. It can either be a small amount of movement or swift, but movement does happen this change in position of an object is called Motion.

If an object is moving, we would be curious to know what are the things happening that make a body move, how long will a body move and many other queries pop in.

To understand the importance of Motion we have a clear example mentioned below:





An object tends to continue in its motion at a constant velocity until and unless an outside force acts on it. The term velocity refers both to the speed and the direction in which an object is moving. It is easy to recognize an object in motion and an object at rest. One must apply an external force to disrupt the balance.

The following are the terms to be recognized before learning Motion:

Rest: When the body does not change its position with respect to the surroundings, the body is said to be at rest.

Motion and Rest are relative terms.

For example: The person sitting inside the moving train is at rest, whereas the person sitting next to him but who is at Motion with the person outside the moving train.

A book on the table is at rest with respect to the table and other objects in the room. But all these objects are sharing the motion of the earth

Or

A car moving on a road is said to be in motion compared to the poles and trees on the roadside. But the people sitting inside the car are at rest compared to one another.


WHAT IS MOTION?

Motion is a change in position of an object or else a process of moving or being moved. When the body changes its position with respect to its surrounding, the body is said to be in Motion.

Examples: Football on ground, motion of moon around earth, rock falling off a cliff, a car moving on the road to trees on the roadside, person inside a moving bus with respect to person outside the bus, bird flying in sky are the examples of motion.

Distance and Displacement

The minimum distance between two points is called displacement while the actual path covered is called distance. The displacement is a vector term and distance is scalar term. Distance and displacement both have SI unit as meter.




AB+BC = distance moved and AC = displacement

The effect of AB+BC is same as effect of AC.

On one round trip, distance is 2(AB+BC) while the displacement = AC+CA = 0

Hence the distance is never zero while the displacement is zero in one round trip.

As we know that the rate of change of displacement is velocity similarly we have,

Speed = Distance moved / Time taken

S = d / t

where d is distance moved.

The SI unit for velocity and speed is meter/second (m/s).

The speed is scalar term and velocity is vector term.

The speed cannot be zero since distance cannot be zero while the velocity can be zero as displacement can be zero.

Types of Motion

The types of motion are: 

1.   Uniform motion
2.   Non uniform motion

a) Uniform motion: When equal distance is covered in equal interval of time, the motion is said to be in uniform motion. The bodies moving with constant speed or velocity have uniform motion or increase at the uniform rate.b) Non Uniform motion: When unequal distances are covered in equal interval of time, the motion is said to be in non uniform motion. The bodies executing non uniform motion have varying speed or velocity.We can even classify motion into three types: 

a.  Translatory motion
b.  Rotatory motion
c.  Vibratory motion

Translatory Motion

In Translatory motion, the particle moves from one point in space to another. This motion may be along a straight line or along a curved path.

They can be classified as: 

(i) Rectilinear Motion: Motion along a straight line is called rectilinear motion. 

(ii) Curvilinear Motion: Motion along a curved path is called curvilinear motion.

Rotatory Motion

In Rotatory motion, the particles of the body describe concentric circles about the axis of motion.

Vibratory Motion

In Vibratory motion, the particles move to and fro about a fixed point.

Equations of Motion

The variable quantities in a uniformly accelerated rectilinear motion are time, speed, distance covered and acceleration. Simple relations exist between these quantities. These relations are expressed in terms of equations called equations of motion.

There are three equations of motion.

Where,

v = Final velocity

u = Initial velocity

a = Acceleration

s = Distance traveled by a body

t = Time taken.






Motion Graphs


a) For body moving at constant velocity:




The graph of straight line parallel to the X axis shows that the body is moving with constant velocity.


b) For uniform motion:

This graph shows the equal displacement in equal interval of time so, the

slope = ΔYΔX gives the change in position over corresponding change in time is constant. Thus, this graph shows the uniform motion.


c) For Body at Rest:



The position-time graph parallel to time axis shows that the body is at rest.


d) For Non uniform motion:




This graph shows unequal distance in equal interval of time which gives the change in position over corresponding change in time which is varying.


Angular Motion

Motion can be angular or uniform. When the body moves on a curved path, there is a change in angular displacement, this is called an angular motion. The rate of change of angular displacement gives angular velocity. It’s a vector term. The angular motion is always an accelerated motion.


Angular velocity (ω) = dθdt

Where dtheta is angular displacement.


Uniform Motion

When the body moves in straight path, equal change in linear displacement in equal interval of time gives uniform motion.

Uniform velocity (v) = dSdt

Where, dS is change in linear displacement

and dt is the time taken.

Speed

Speed = Distance moved / Time taken

S = d / t

where, d is distance moved.


The SI unit for velocity and speed is meter/second (m/s). The speed is scalar term and velocity is vector term. The speed cannot be zero since distance cannot be zero while the velocity can be zero as displacement can be zero.




Newton's Law of Motion

Newton has given the three laws of motion.Newton’s First law of motion: The body remains in rest or in continue motion unless some external force is applied on it.Example: The book on table will remain on table unless some force is applied on it. The ball moving on ground stops by itself because of friction (external force). If there were no frictional forces, the moving ball will continue to move unless we stop it.



First law of motion is related to term “Inertia”. It’s the property of body by the virtue of which the body resists the external force.


Common examples of inertia in our day to day life: 

The passengers fall forward when the bus suddenly stops. This is due to inertia of motion, the lower portion of body comes to rest but the upper portion of body continue to be in motion. 

When we shake the branches, the fruits and leaves fall. The branches are in motion while the fruits and leaves are in rest so, they fall. 

The dust particles get removed when we shake the carpet. This is, because the particles are at rest while the carpet is moving, so, the particles are removed. 

When the person jumps from the moving bus, he runs through some distance due to inertia of motion. 

Any moving body has momentum. Mathematically, the momentum is denoted by P. It’s the product of mass and its velocity.

P = mass x velocity
P = m x v

Newton’s Second law of motion: According to the Newton’s second law of motion the rate of change of momentum is directly proportional to the force applied and acts in the direction of force.

F = dPdt ……………………………….(1)

dP is change in momentum


F = d (mv–mu)dt……………………(2)

m = mass of body.


F = m d(v−u)dt


∴ F = ma …………………….(3)

a = v–ut

Force = mass x acceleration

Hence, a = Fm

For constant force, acceleration produced in the body, is inversely proportional to the mass of the body. Larger is mass, lesser is acceleration produced.

For equal masses of the body, the acceleration is directly proportional to the force applied. Larger is force, higher is acceleration produced.

Newton’s Third law of motion: To every action, there is an equal and opposite reaction.

Common examples of Newton’s third law in our day today life: 

When a person jumps from a boat, the boat moves backwards. 

When a bullet is fired, the gun goes backwards. 

The huge amount of smoke downward, pushes the rocket upwards. 

When a balloon is blown, the air rushes outward while the balloon moves backward with the same momentum