1. MOTION - TopperLearning

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1. MOTION

1. Physical Science

This science deals with the properties and behavior of non-living things.

Physics: It is the branch of science which deals with the study of the natural laws and their manifestation in the natural phenomenon.

Mechanics (oldest branch): It deals with the conditions of rest or motion of the material objects around us.

Statics: It deals with the study of objects at rest or in equilibrium, even when they are under the action of several forces (measurement of time is not essential).

Kinematics: It deals with the study of motion of objects without considering the cause of motion (measurement of time is essential). [Kinematics : Greek word Kinema → motion]Dynamics: It deals with the study of objects taking into consideration the causes of their motion.

[Dynamics : Greek word Dynamis → power]

2. Rest and Motion

Rest: An object is said to be at rest if it does not change its position with respect to its surroundings, with the passage of time.

In the above definition, surrounding means one or more stationary objects around the given body. These stationary objects can be called reference points.

For example, if a chair is placed in the middle of the room, then the walls of the room are its surroundings. If the position of the chair (i.e., its distances from the walls) does not change with respect to the walls as well as time, then the chair is at rest.

1.2 Motion

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Motion: A body is said to be in motion if its position changes continuously with respect to the surroundings (or with respect to an observer) with the passage of time.

In other words, when the position of a body continuously changes with respect to the reference point (any object around the body) and time, the body is said to be in a state of motion.

PLANCESS CONCEPTS

Motion is a combined property of the object under study and the observer. There is no meaning of rest or motion without the viewer. Nothing is in absolute rest or in absolute motion.

Vipul Singh

AIR 1, NTSE 2009

2.1 Rest and Motion are Relative Terms

Apparently rest and motion are the terms which are opposite to each other. However, sometimes it becomes very difficult to say whether a body is in the state of rest or state of motion.

Consider a person sitting in a speeding train. If the person compares himself with respect to fellow passengers and the interiors of the train, he finds that his position does not change with respect to his immediate surroundings and hence, he is in the state of rest.

However, if the person compares himself with respect to the trees, fields, etc., outside the train, he finds that his position rapidly changes with respect to the outside surroundings and hence, he is in the state of motion.

As a matter of fact, if the immediate surroundings are limited (small), then a body may be in the state of rest. However, if the surroundings are enlarged, the body may be in the state of motion. It is for the same reason that we find ourselves in the state of rest on the planet Earth with respect to our immediate surroundings. However, we are in the state of motion, if we compare ourselves with respect to the Moon, the Sun or the Stars.

Example: A, B and C are three persons. B and C are sitting in the car and A is standing outside it. When car starts to move, B and C are changing their position with respect to A so B and C are in motion with respect to A but B is not changing its position with time with respect to C, so B is at rest with respect to C (same for C). Therefore motion depends on the position of the observer, hence motion is relative.

Example: We know that the earth is rotating about its axis and revolving around the sun. The stationary objects like your classroom, a tree and the lamp posts etc., do not change their position with respect to each other i.e. they are at rest. Although earth is in motion. To an observer situated outside the earth, say in a space ship, our classroom, trees etc. would appear to be in motion. Therefore, all motions are relative. There is nothing like absolute motion.

1.3 Foundation for Physics

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2.2 Concept of a Point Object

In mechanics while studying the motion of an object, sometimes its dimensions are of no importance and the object may be treated as a point object without much error. When the size of the object is much less in comparison to the distance covered by the object then the object is considered as a point object.

Example: If one travels by a car from one place to another faraway place, then length of the car is ignored as compared to distance traveled.

Example: Earth can be regarded as a point object for studying its motion around the sun.

2.3 Frame of Reference

To locate the position of an object we need a frame of reference. A convenient way to set up a frame of reference is to choose three mutually perpendicular axes and name them x-y-z axes. The coordinates (x, y, z) of the particle then specify the position of object with respect to this frame. If any one or more coordinates change with time, then we say that the object is moving with respect to this frame.

PLANCESS CONCEPTS

There is no rule or restriction on the choice of a frame. We can choose a frame of reference according to our convenience to describe the situation under study. If two frames are moving with respect to each other with uniform velocity, acceleration of a body is same in both the frames.

Neeraj Toshniwal

AIR 23 , NSO

2.4 Characteristics of a Moving Body

1. There must be a reference point (stationary object) to describe the position of the given body.

2. The position of the given body must continuously change with time and with respect to the reference point.

2.5 Motions in One, Two and Three Dimensions (Type of Motion)

As position of the object may change with time due to change in one or two or all the three coordinates, so we have classified motion as follows:

1.4 Motion

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2.5.1 Motion in 1-D

If only one of the three coordinates specifying the position of object changes with respect to time. In such a case the object moves along a straight line and the motion therefore is also known as rectilinear or linear motion.

Example: (i) Motion of train along straight railway track.

(ii) An object falling freely under gravity.

(iii) When a particle moves from P1 to P2 (figure 1.1) along a straight line path only the x-coordinates changes.

2.5.2 Motion in 2-D

If two of the three coordinates specifying the position of object changes with respect to time, then the motion of object is called two dimensional. In such a motion the object moves in a plane.

Example: (i) Motion of queen on carom board.

(ii) An insect crawling in a zigzag path on the floor of the room.

(iii) Motion of object in horizontal and vertical circles etc.

(iv) Motion of planets around the sun.

(v) A car moving along a zigzag path on a level road.

2.5.3 Motion in 3-D

If all the three coordinates specifying the position of object changes with respect to time, then the motion of object is called 3-D. In such a motion, the object moves in a space.

Example: (i) A bird flying in the sky (also kite).

(ii) Random motion of gas molecules.

(iii) Motion of an aeroplane in space.

Figure 1.1: Straight line motion


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PLANCESS CONCEPTS

Reference Frames:

In order to agree upon the location of an object, we have to establish a common reference frame. A reference frame consists of the reference object and the reference directions used in our description. When we say that canteen is north of the lecture Hall, the reference frame consists of canteen and the compass direction north. Once we define our reference frame, others should be able to place themselves in that same reference frame and agree upon the location of canteen. While the strict direction of reference frame requires both a reference object and a set of reference directions, ordinarily we just mention the reference object. “It’s on the table” and “she’s at the office” both identify reference objects- the table and the office. The terms on and at, however, imply a set of commonly agreed-upon reference directions. This works fine as long as we are all in the same earth-based reference frame.

Shivam Agarwal

AIR 3, NSO

3. Types of Motion

In our everyday life, we come across various types of motion, like the motion of the sun, the motion of the wind, etc.

Motion can be classified into:

1. Random motion

2. Translational motion

3. Rotational motion

4. Oscillatory or vibratory motion

3.1 Random Motion

In this type of motion, the particles move randomly (they do not move along a definite path). Example: The motion of dust particles in wind or in air.

3.2 Translational Motion

In this type of motion, every particle of the body has the same displacement. Translational motion can be along a straight line or along a curved path. The motion along a straight line is called rectilinear motion and the motion along a curved path is called curvilinear motion.

1.6 Motion

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Example:

Rectilinear Motion: Train speeding along a straight track. It is also referred to as one dimensional motion.

Curvilinear Motion: The path of a ball hit by a batsman for a six. This type of motion is also referred to as a two or three dimensional motion.

3.3 Rotational Motion

If the particles of the body revolve in a circle about the same axis, then the motion is said to be rotational.

Examples:

(i) Rotation of Earth on its axis.

(ii) Pulley used in drawing water.

3.4 Oscillatory or Vibratory Motion

Any to and fro motion about a fixed point is called oscillatory or vibratory motion.

(i) When the string of a guitar is plucked, it performs an oscillatory motion.

(ii) The motion of the pendulum of a clock.

(iii) The motion of a simple pendulum, a body suspended from a string.

4. Scalar and Vector Quantities

Physical quantities (i.e. quantities of physics) can be divided into two types:

(i) Scalar quantity: Any physical quantity, which can be completely specified by its magnitude alone, is a scalar quantity or a scalar.

Example: Charge, mass, distance, area, speed, time, temperature, density, volume, work, power, energy, potential etc.

(ii) Vector quantity: Any physical quantity, which requires direction in addition to its magnitude is known as a vector.

Example: Displacement, velocity, acceleration, force, momentum, weight (force) and electric field etc.


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4.1 Representation of a Vector

A vector is represented by a directed line segment drawn in the given direction on a certain scale.

Tail head (symbolic representation)

Example 1: To represent a displacement of 25 m towards east. (Scale 5 m = 1 cm on graph paper.)

Example 2: To represent a velocity of 20 km/h towards 30° east of south. (Scale 5 km/h = 1 cm on graph paper.)

Example 3: 6 m displacement, 60° north - east (north of east) (Scale 1 m = 1 cm on graph paper)

1.8 Motion

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4.2 Difference between Scalar and VectorTable 1.1: Properties of Scalar and Vector quantity

Scalar Vector

1. They have a magnitude only. 1. They have magnitude as well as direction.

2. They are added or subtracted arithmetically like 3 kg + 5 kg = 8 kg

2. They are added or subtracted by the process of vector addition.

5. Distance and Displacement

5.1 Distance

Consider a body traveling from A to B along any path between A and B. The actual length of the path that a body travels between A and B is known as the distance. The distance traveled is different for different paths between A and B. It is a scalar quantity. According to figure 1.2 distance of path APB is AP + PB and of path AB is AB.

S.I. unit of distance is meter.

5.2 Displacement

The distance traveled in a given direction is the displacement. Thus displacement is the shortest distance between the given points. It is a vector quantity. S.I. unit of displacement is meter.

NOTE: If a body travels in such a way that it comes back to its starting position, then the displacement is zero. However, distance traveled is never zero.

Example:

(i) When an object moves towards right from origin, its displacement is positive (figure 1.3).

(ii) When an object moves towards left, its displacement is negative.

(iii) When an object remains stationary or it moves first towards right and then an equal distance towards left, its displacement is zero.

(iv) Shift of origin to new point causes no change in displacement.

Figure 1.2: Different paths of the particle

Figure 1.3: Negative and positive displacement of the particle

X�

O AX�

A O

X is positive�

X is negative�


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PLANCESS CONCEPTS

Displacement is a vector quantity. The magnitude of the displacement is the length of the straight line joining the initial and final position. The direction is from the initial to the final position.

Shivam Agarwal

AIR 3, NSO

5.3 Difference between Distance and Displacement

Suppose a student lives at a place A and he has to reach his school at another place C, but first he has to give company to his friend living at place B. The student starts from point A and travels a distance of 10 km to reach B and then travels another distance of 6 km from B to reach C. Thus, the student goes along the path ABC (shown by dotted lines). Length of the path ABC gives the actual distance travelled by the student (figure 1.4).

Thus, the distance travelled by a body is the actual length of the path covered by a moving body irrespective of the direction in which the body travels.

In this case, the actual length of the path covered by man is 10 km + 6 km = 16 km, so the distance travelled by the student is 16 km.

Another aspect of this event is that, when the student has reached point C, the student is AC distance away from point A and this is the shortest distance between point A and C. We can say, AC is the shortest distance between the starting point and the destination point. The length of line AC is the shortest distance between two points which is also called the magnitude of displacement.

When a body moves from one position to another, the shortest (straight line) distance between the initial position and final position of the body, along with direction is known as its displacement.

In the above illustration, the shortest distance between the initial position A and final position C of the student is 8 km, so the displacement of the student is 8 km in the east direction.

Thus, distance is a scalar quantity (because it has magnitude or numerical value only but displacement is a vector quantity (because it has magnitude as well as direction).

Figure 1.4: Distance and Displacement

1.10 Motion

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Table 1.2: Differences between distance and displacement

Distance Displacement

1 It is defined as the actual path traversed by a body.

It is the shortest distance between the two points between which the body moves.

2 It is a scalar quantity. It is a vector quantity.

3 It can never be negative or zero. It can be negative, zero or positive.

4 Distance can be equal to or greater than displacement.

Displacement can be equal to or less than distance.

5 Distance travelled is not a unique path between two points.

Displacement is a unique path between two points.

6 Distance never decreases with time and for a moving body it is never zero.

Displacement can decrease with time and for a moving body it can be zero.

7 Distance in SI is measured in meter. Displacement in SI is measured in meter.

Illustration 1: A body is moving along a circular path of radius r. What will be the distance and displacement of the body when it completes half a revolution?

Solution: Distance = length of arc = πr

Displacement = shortest straight line distance between points A and B = 2r

Illustration 2: A body moves 4m towards east and then 3m north. What are the displacement and distance covered by the body?

Solution: Distance = AB + BC = (4 + 3) m = 7m

Displacement is

2 2AC AB BC 16 9 5m= + = + =


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6. Uniform and Non-Uniform Motion

6.1 Uniform Motion

A body has a uniform motion if it travels equal distances in equal intervals of time, no matter how small these time intervals may be. For example, a car running at a constant speed of say, 10 meters per second, will cover equal distances of 10 meters every second, so its motion will be uniform. Please note that the distance-time graph for uniform motion is a straight line (as shown in the figure 1.5).

6.2 Non-Uniform Motion

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 building, we will find that it covers unequal distances in equal intervals of time.

It covers:

4.9 meters in the 1st second,

14.7 meters in the 2nd second,

24.5 meters in the 3rd second, and so on.

Thus, a freely falling ball covers smaller distance in the initial ‘1 second’ interval and larger distance in the later ‘1 second’ interval. From this discussion we conclude that the motion of a freely falling body is an example of non-uniform motion. The motions of a train starting from the railway station is also an example of non-uniform motion. This is because when the train starts from a station, it moves a very small distance in the ‘first’ second. The train moves a little more distance in the ‘2nd’ second and so on. And when the train approaches the next station, the distance traveled by it per second decreases.

Please note that the distance-time graph for a body having non-uniform motion is curved line (as shown in the figure 1.6). Thus, in order to find out whether a body has uniform motion or non-uniform motion, we should draw the distance-time graph for it. If the distance time graph is straight line, the motion will be uniform and if the distance-time graph is a curved line, the motion will be non-uniform. It should be noted that non-uniform motion is also called accelerated motion.

7. Speed

The distance traveled by a body in unit time is called its speed. Therefore,

Speed = DistanceTime

or s = dt

. S.I. unit of speed or average speed is m/sec. It is a scalar quantity.

Figure 1.5: Distance Vs Time of uniform motion

Figure 1.6: Distance Vs Time of non-uniform motion

1.12 Motion

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The speed of a moving body is a quantity that tells us how fast the body is moving.

7.1 Average Speed

The overall rapidity of an object in a given time interval is decided by a quantity called average speed. It is the total distance travelled by the object divided by the total time taken to cover that distance.

Average speed = total dis tance travelled

total time taken

Thus the average speed of an object in a time interval is equal to the distance traversed divided by the time interval.If a body travels a distance s in time t, then its speed v is given by v = s/t.The SI unit of distance is meter (m) and that of time is second (s), therefore, the SI unit of speed is meter per second which is written as m/s or ms–1. Speed has only magnitude, it has no specified direction, and therefore, speed is a scalar quantity.Example while travelling in a car you cannot maintain constant speed because of the traffic and red light, you have to slow down or stop the car. Thus, the speed of car is usually not constant. For illustration, if a car takes 3 hrs to travel 75 km, the average speed of the car is 25 km

per hour. Although the average

speed of car is 25 km per hour, it does not mean that the car is moving at this speed for all the time.

7.2 Uniform Speed (or Constant Speed)

A body has a uniform speed if it travels equal distance in equal intervals of time, no matter how small this time interval may be. For illustration, if you are driving motor bike with a uniform speed of 40 km per hour, it means you are travelling 20 km in every 30 minutes, or you are covering 10 km in every 15 minutes.

7.3 Variable Speed (Non-Uniform Speed)

If a body covers unequal distances in equal intervals of time, its motion is said to be non-uniform.

Example Falling of an apple from a tree, a cyclist moving on a rough road, an athlete running a race, vehicle starting from rest, the motion of freely falling body, etc.

7.4 Instantaneous Speed

The speed of an object at any particular instant of time or at a particular point of its path is called the instantaneous speed of the object. It is measured by speedometer in an automobile.

Illustration 3: A car travels 30 km at a uniform speed of 40 km/h and the next 30 km at a uniform speed of 20 km/h. Find its average speed.

Solution: We know that, dis tanceSpeedtime

=


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So, let the time taken to cover first 30 km be t1, = =130 3

t hours hours40 4

And let the time taken to cover next 30 km be t2, = =230 3

t hours hours20 2

So, the total time 3 3 94 2 4

= + = hours

Average speed = Total dis tance 60 460Total time 9 / 4 9

×=

Average speed = 26.6 km/hr.

Illustration 4: On a 120 km track, a train travels the first 30 km with a speed of 30 km/hr. How fast must the train travel the next 90 km so as to get average speed of 60 km/hr for the entire trip?

Solution: Let the speed of the train be u km/h for next 90 km. Then,

1

90tu

= where t1 is time taken to cover 90 km.

and time taken to cover first 30 km is say t2, 230t 1hr30

= = .

So, total time = 1 290t t 1u

+ = +

Total dis tanceAverage speed

Total time=

12060 90 1u

⇒ =+

; ⇒ u = 90 km/hr.

8. Velocity

It is the rate of change of displacement.

Therefore, velocity = displacementtime

or it is the distance traveled in unit time in a given direction.

Velocity = dis tance travelled in a given direction

time taken

S.I. unit of velocity is m/s. It is a vector quantity.

(Magnitude of the velocity is known as speed)

1.14 Motion

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Speed of a body gives us the total path travelled by a body in given time while velocity of a body is defined as “rate of change of displacement”. Since the change of position of a body may be in any direction viz north, south, etc., so for change of position you have to define the direction also.For illustration, a car moves 5 km east, 2 km north, 3 km east and then 2 km south. Now, the final position of the car is 8 km east which is shown in the given diagram.

In figure 1.7, the car makes its journey from point A to B (which is 5 km east), from B to C (which is 2 km north), C to D (which is 3 km east), and then from D to E (which is 2 km south) then it reaches at a point E which is 8 km from A towards east direction. So, we can say that the change of position is 8 km from A towards east, which is also known as displacement. But if we consider the total path travelled by car which is 5 + 2 + 3 + 2 = 12 km it is called the distance. So, the velocity can be defined as “rate of change of displacement”.

change of position (displacement)Velocity

time=

In the previous illustration, if total time taken by the body is 4 hours, then the velocity of body can be

said to be 8km towards eastVelocity

4hrs=

Velocity = 2 km/hr towards east

NOTE:

To covert speed/velocity from m/sec to km/hr, multiply it by 18/5.

To covert speed/velocity from km/hr to m/sec, multiply it by 5/18

Velocity is written by v and the SI unit of velocity is meter per second or m/s or ms–1. Since we required direction to define velocity, therefore velocity is a vector quantity.

Table 1.3: Differences between Speed and Velocity

Speed Velocity

1. It is a scalar quantity.

2. Speed = dis tance travelled

time3. It is the rate of change of distance along the

path of an object.

1. It is a vector quantity.

2. Velocity = displacementtime

3. It is the rate of change of position of an object in specific direction.

Figure 1.7: Path followed by the particle


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8.1 Uniform Velocity (Constant Velocity)

If the magnitude as well as direction of velocity do not change with time, then we say that velocity is uniform. Thus, body has a uniform velocity if it travels in a straight line and moves equal distances in equal intervals of time, no matter how small these time intervals may be.

The velocity of a body can be changed in three ways:

(i) By changing the speed of the body,

(ii) By keeping the speed constant but changing the direction

(iii) By changing the speed as well as direction.

Suppose a car is moving on a circular road with constant speed. Now, though the speed of the car is constant, its velocity is not uniform because the direction of car is changing continuously.

Illustration 5: A body travels with velocity v1 for time t1 and with velocity v2 for time t2 in the same direction. Find the average velocity of the body.

Solution: Displacement in time (t1 + t2) = v1t1 + v2t1

∴ Average velocity = 1 1 2 2

1 2

v t v tdisplacementtotal time taken t t

+= =

+

Illustration 6: A body traveling along a straight line traversed one third of the total distance with a velocity 4 m/s. The remaining part of the distance was covered with a velocity 2 m/s for half the time and with velocity 6 m/s for the other half of time. What is the average velocity over the whole motion?

Solution: Let S be the total distance travelled by the body. If t1 is the time taken to cover first one third distance,

Then, 1s / 3 s

t4 12

= =

Let t2 be the time for each of the remaining two journeys. Then

2 2 2

2s2t 6t 8t

3= + = ; 2

2s st

3 8 12⇒ = =

×

displacementAverage velocity

total time∴ = = ( ) ( )1 2

s s 4m /st 2t s /12 2 s /12

= =+ +

Illustration 7: In one second a particle goes from A to a point B moving in a semicircle (see figure1.13). What is the magnitude of average velocity?

Solution: displacement (AB)Average velocity

time= 2r

2r m / s1

= =

1.16 Motion

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Illustration 8: A lecture theatre of a college is 40 ft. wide and a door is situated at a corner. A teacher enters at 12.00 noon through the door and makes 10 rounds along the 40 ft. wall back and forth during the period and finally leaves the class room at 12.50 pm. through the same door. Compute his average speed and average velocity.

Solution: Total distance travelled in 50 minutes = 800 ft.

Average speed =800

50ft/min= 16 ft/min.

At 12.00 noon he is at the door and at 12.50 he is again at the same door.

Therefore displacement during the 50 min interval is zero.

∴ Average velocity = zero.

8.2 Non-Uniform Velocity

When the velocity of a body is changing with time, the body is said to be in non-uniform motion. When the speed of a body is decreasing or increasing or the direction of motion of the body is changing, the body is said to be in non-uniform motion. Non-uniform motion can be divided in two categories. One is called accelerated and the other is called retarded motion. If speed is constant then also body can have a non-uniform velocity. Example: A car moving on a circular road with constant speed.

PLANCESS CONCEPTS

For motion along a curved path, the direction of the instantaneous velocity v of a particle is

always tangent to the particle’s path at the particle’s position.

Anand K

AIR 1, NSO 2011

8.3 Average Velocity

If initial velocity of body is u and final velocity is v then the arithmetic means of velocity is called

average velocity and is given as avu v

v2+

= . Also for an object moving with variable velocity it is

defined as the ratio of its total displacement to the total time interval in which the displacement occurs.

Average velocity = Total displacement

Total time. If x1 and x2 are the positions of an object at times t1 and t2 then,

2 1av

x x xv

t t

− ∆= =

∆ ∆

∆ = − ∆ = −

x x x ; t t t2 1 2 1


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8.4 Instantaneous Velocity

The velocity of an object at any given instant of time at particular point of its path is called its instantaneous velocity.

t 0

x dxV lim

t dt∆ →

∆= =

∆

Illustration 9: When is the average speed of an object equal to the magnitude of its average velocity? Give reason also.

Solution: As average speed = total path lengthtime int erval

also, average velocity = DisplacementTime int erval

. When an object

moves along a straight line and in the same direction its total path length is equal to the magnitude of its

displacement. Hence average speed is equal to the magnitude of its average velocity.

8.5 Features of Velocity in Uniform Motion

(i) The velocity in uniform motion does not depend on the choice of origin.

(ii) The velocity in uniform motion does not depend on the choice of the time interval (t2 – t1).

(iii) For uniform motion along a straight line in the same direction, the magnitude of the displacement is equal to the actual distance covered by the object.

(iv) The velocity is positive if the object is moving towards the right of the origin and negative if the object is moving towards the left of the origin.

(v) For an object in uniform motion the net external force in the object is ZERO.

(vi) In uniform motion, the instantaneous velocity is equal to the average velocity at all times because velocity remains constant at each instant, at each point of the path.

Illustration 10: A car is moving along x-axis. As shown in the figure it moves from O to P in 18 s and returns from P to Q in 6 second. What is the average velocity and average speed of the car in going from (i) O to P and (ii) from O to P and back to Q?

Solution: (i) Average velocity = Average speed = path length 360m

20 m / stime int erval 18s

= =

1.18 Motion

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(ii) From O to P and back to Q:

Average velocity = 240mOQ 10 m / s

18 6 24s= =

+

Average speed = path length OP PQ 360 120

20 m / stime int erval 18 6 24

+ += = =

+

Illustration 11: A car covers the 1st half of the distance between two places at a speed of 40 km hr-1 and the 2nd half at 60 km/h. What is the average speed of the car?

Solution: Suppose the total distance covered is 2S.

Then time taken to cover first distance with speed 40 km/hr,

1

St hr40

=

Time taken to cover second S distance with speed 60 km/hr,

2St hr60

= ; ( )avtotal disance S S

Vtotal time S / 40 S / 60

+= =

+

av2s 2sV 120

(3s 2s ) /120 5s= = ×

+ ; ⇒ Vav = 48 km/hr

Illustration 12: A non-stop bus goes from one stop to another stop with a speed of 54 km/hr, and the same bus returns from the second stop to the first station with a stop of 36 km/hr. Find the average speed of the bus for the entire journey.

Solution: Suppose the distance between the stations is S. Time taken in reaching from first station to second station.

1

St h54

=

Time taken while returning back, 2St h36

=

Total t = t1 + t2 ; 2S 3SS S 5St h

54 36 108 108+

= + = =


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Average speed av

Total dis tanceV

Total time=

av av

25 216V 108; V 43.2 km / h55 5

= × = =

9. Acceleration

Mostly the velocity of a moving object changes either in magnitude or in direction or in both when the object moves. The body is then said to have acceleration. So the rate of change of velocity i.e. change in velocity in unit time is equal to the acceleration (it is a vector quantity). Its S.I. unit is ms-2 and c.g.s unit is cms-2.

Acceleration =

change in velocity final velocity initial velocityv utime t time

−−= =

Suppose a bus starts its journey from rest (initial velocity is zero) in a straight line and its velocity increases at a steady rate so that after 10 seconds its velocity is 10 meters per second. Now, in 10 seconds, the velocity has increased by 10 – 0 = 10 meters per second and in one second the velocity increases by 10/10 = 1 m/s. In other words, we can say the velocity increases 1 meter per second every second. The bus is said to have an acceleration of 1 meter per second per second or 1 m/s2 or 1 ms–2.

The acceleration has a direction in which the change in velocity takes place. The acceleration is taken to be positive if it is in the direction of the velocity, and negative when opposite to the direction of velocity. Negative acceleration is called retardation.

9.1 Uniform Acceleration (Uniformly Accelerated Motion)

If a body travels in a straight line and its velocity increases in equal amounts in equal intervals of time, then its motion is known as uniformly accelerated motion.

Example 1: Motion of a freely falling body is an example of uniformly accelerated motion (or motion of a body under the gravitational pull of the earth).

Example 2: Motion of a bicycle going down the slope of a road when the rider is not pedaling and wind resistance is negligible.

9.2 Non-Uniform Acceleration

If during motion of a body its velocity increases by unequal amounts in equal intervals of time, then its motion is known as non-uniform accelerated motion.

Example 1: Car moving in a crowded street. The speed of a car running on a crowded city road changes continuously. At one moment, the velocity of car increase whereas at another moment

1.20 Motion

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it decrease. So, in every small interval of time, the velocity changes non-uniformly. So, the acceleration is non-uniform.

Example 2: Motion of a train leaving or entering the platform.

PLANCESS CONCEPTS

When a particle is moving in a curved path, it always has nonzero acceleration, even when it moves with constant speed. There is a non-zero acceleration whenever the velocity vector changes in any way, whether there is a change of speed, direction, or both. The vector

a points towards the concave side of the path - that is, towards the inside of any turn that the particle is making.

Vaibhav Gupta

AIR 2, NSO

9.3 Types of Acceleration

(i) Positive acceleration: If the velocity of an object increases in the same direction as the original one, the object has a positive acceleration.

(ii) Negative acceleration (retardation): If the velocity of a object decreases in the same direction as the original one, the body has negative acceleration or it is said to be retarding.

9.4 Equations of Uniformly Accelerated Motion

9.4.1 First Equation of Motion

Consider a body having initial velocity ‘u’. Suppose it is subjected to a uniform acceleration ‘a’ so that after time ‘t’ its final velocity becomes ‘v’. Now we know,

Acceleration = change in velocitytime

; −

=v u

at

or v = u + at or v = at + u ……(i)

9.4.2 Second Equation of Motion

Suppose a body has an initial velocity ‘u’ and uniform acceleration ‘a’ for time ‘t’ so that its final velocity becomes ‘v’. The distance traveled by moving body in time ‘t’ is ‘s’ then the average velocity = (v + u)/2. Distance traveled = Average velocity × time

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