introduction to motion assignment

21 Introduction to Motion

Objects are in motion everywhere we look. Everything from a tennis game to a space-probe flyby of the planet Neptune involves motion. When you are resting, your heart moves blood through your veins. And even in inanimate objects, there is continuous motion in the vibrations of atoms and molecules. Questions about motion are interesting in and of themselves: How long will it take for a space probe to get to Mars? Where will a football land if it is thrown at a certain angle? But an understanding of motion is also key to understanding other concepts in physics. An understanding of acceleration, for example, is crucial to the study of force.

Our formal study of physics begins with kinematics which is defined as the study of motion without considering its causes . The word “kinematics” comes from a Greek term meaning motion and is related to other English words such as “cinema” (movies) and “kinesiology” (the study of human motion). For now, we will study only the motion of a football, for example, without worrying about what forces cause or change its motion. Such considerations come in other chapters. In this chapter, we examine the simplest type of motion—namely, motion along a straight line, or one-dimensional motion. Later, we apply concepts developed here to study motion along curved paths (two- and three-dimensional motion); for example, that of a car rounding a curve.

Introduction to Motion by OpenStax is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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• Introduction To Motion

Introduction to Motion

When we talk about motion or rest, it is with reference to some point known as the origin. So, now with respect to the change in the position, we have two quantities which can be used to describe that change in position. They are distance and displacement. So now the question is, what is the difference between the two? Talking about distance is defined as the total path length covered during the motion. It can be represented only by magnitude. On the other hand, displacement is the shortest distance between the initial and final positions. It requires both magnitude and direction for complete representation.

What Is Motion?

We can define motion as the change of position of an object with respect to time. A book falling off a table, water flowing from the tap, rattling windows, etc., all exhibit motion. Even the air that we breathe exhibits motion! Everything in the universe moves. We live in a universe that is in continual motion. The fundamental particle of matter, that is, the atom, is in constant motion too. Every physical process in the universe is composed of motion of some sort. The motion can either be swift or slow, but motion exists. It is important that we give due attention to the study of motion because of its importance in the physical world. Motion is mainly described in terms of the following terms:

• Displacement

So now that we have a basic idea of both, we will try to solve an example: Suppose the distance between two cities, A and B, is ‘d’. A person goes from A to B and returns. Calculate distance travelled and displacement.

Distance travelled = Total path length covered

Displacement is measured as the shortest distance between the initial and final position. In this case, both are the same, and hence, displacement is also zero.

So for a motion, can displacement be greater than the distance covered? Think about it, and if not, can it be equal?

You may also want to check out these topics given below!

• Force And Motion
• Change In State Of Motion
• Newton’s Laws of Motion
• Motion In Physics

The below video provides the Top 10 NTSE Important Questions on Motion Class 9

Types Of Motion

We might have noticed that different objects move differently. Some objects move in a curved path, some in a straight path and a few others in a different way. According to the nature of the movement, motion is classified into three types as follows:

• Linear Motion
• Rotary Motion
• Oscillatory Motion

In linear motion, the particles move from one point to another in either a straight line or a curved path. The linear motion depending on the path of motion, is further divided as follows

• Rectilinear Motion – The path of the motion is a straight line.
• Curvilinear Motion – The path of the motion is curved.

A few examples of linear motion are the motion of the train, football, the motion of a car on the road, etc.

Rotatory Motion

Rotatory motion is the motion that occurs when a body rotates on its own axis.  A few examples of the rotatory motion are as follows:

• The motion of the earth about its own axis around the sun is an example of rotary motion.
• While driving a car, the motion of wheels and the steering wheel about its own axis is an example of rotatory motion.

Oscillatory motion is the motion of a body about its mean position. A few examples of   oscillatory motion are

• When a child on a swing is pushed, the swing moves to and fro about its mean position.
• The pendulum of a clock exhibits oscillatory motion as it moves to and fro about its mean position.
• The string of the guitar when strummed moves to and fro by its mean position resulting in an oscillatory motion.

Watch the Video and Learn about the Basic Concepts of Motion

Examples of Motion

Now let us understand motion clearly with the help of a few examples.

• Our daily activities, like walking, running, closing the door, etc. involve motion. There is a change of position of the object involved in these activities.
• The flow of air in and out of our lungs is also an example of motion.
• The automobiles that carry passengers from the place of pick up to the destination possess motion. In this case, the position of passengers is changed from one place to another.

Frequently Asked Questions – FAQs

State motion definition, what are the types of motion.

The following are the types of motion:

What are the types of linear motion?

• Rectilinear Motion
• Curvilinear Motion

State true or false: Displacement is measured as the shortest distance between the initial and final position.

Give some examples of motion.

Examples of motion are:

• The flow of air in and out of our lungs.
• The automobiles that carry passengers from the place of pick up to the destination.

Stay tuned with BYJU’S to learn more about other Physics concepts with the help of interesting video lessons.

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2: Motion in One Dimension

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This lesson is based on Chapter 2 (2.1-2.7) and Chapter 4 (4.1-4.4) and Chapter 6 (6.5) of the OpenStax College Physics textbook. Each section of the chapter will include my lecture slides as well as an embedded link to my YouTube video for that lecture.

Browse Course Material

Course info, instructors.

• Prof. Deepto Chakrabarty
• Dr. Peter Dourmashkin
• Dr. Michelle Tomasik
• Prof. Anna Frebel
• Prof. Vladan Vuletic

Departments

As taught in.

• Classical Mechanics

Learning Resource Types

Week 10: rotational motion.

« Previous | Next »

• Week 10 Introduction

Lesson 28: Motion of a Rigid Body

• 28.1 Rigid Bodies
• 28.2 Introduction to Translation and Rotation
• 28.3 Review of Angular Velocity and Acceleration

Lesson 29: Moment of Inertia

• 29.1 Kinetic Energy of Rotation
• 29.2 Moment of Inertia of a Rod
• 29.3 Moment of Inertia of a Disc
• 29.4 Parallel Axis Theorem
• 29.5 Moment of Inertia of a Sphere
• 29.6 Derivation of the Parallel Axis Theorem

Lesson 30: Torque

• 30.1 Introduction to Torque and Rotational Dynamics
• 30.2 Cross Product
• 30.3 Cross Product in Cartesian Coordinates
• 30.4 Torque
• 30.5 Torque from Gravity

Lesson 31: Rotational Dynamics

• 31.1 Relationship between Torque and Angular Acceleration
• 31.2 Internal Torques Cancel in Pairs
• 31.3 Worked Example - Find the Moment of Inertia of a Disc from a Falling Mass
• 31.4 Worked Example - Atwood Machine
• 31.5 Massive Pulley Problems
• 31.6 Non-Slip Condition
• 31.7 Worked Example - Two Blocks and a Pulley Using Energy

Week 10 Worked Example

• PS.10.1 Blocks and Massive Pulley

Week 10 Problem Set

• Problem Set 10

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Introduction to Motion in One Dimension

A certain amount of matter limited in all directions and consequently having a finite size, shape and occupying some definite space is called a body.

A particle is defined as a portion of matter infinitesimally small in size so that for the purpose of investigation, the distance between its different parts may be neglected. Thus, a particle has only a definite position, but no dimension. In the problems we are going to discuss, we will consider a body to be a particle for the sake of simplicity.

The position of object can change on a straight line (like on x-axis with respect to origin) or on a plane with respect to some fixed point on frame. So we can define motion as follows:-

An object or a body is said to be in motion if its position continuously changes with time with reference to a fixed point (or fixed frame of reference).

But note that, the moving object is either a particle, a point object (such as an electron) or an object that moves like a particle. A body is said to be moving like if every portion of it moves in the same direction and at the same rate.

Motion in One Dimension

When the position of object changes on a straight line i.e. motion of object along straight line is called motion in one dimension.

To understand the essential concepts of one dimensional motion we have to go through some basic definitions.

Frame of Reference

A frame of reference is a set of coordinate axes which is fixed with respect to a space point (a body or an object can also be treated as a point mass therefore it can become a site for fixing a reference frame), which we have arbitrarily chosen as per our observer's requirement. The essential requirement for a frame of reference is that, it should be rigid.

Position of an Object 

  The position of an object is defined with respect to some frame of reference. As a convention, we define position of a point (essentially we treat body as a point mass) with the help of three co-ordinates X, Y and Z. Hence X, Y, Z is a set of coordinate axes representing a 3-dimensional space and each point in this space can be uniquely defined with the help of a set of X, Y and Z coordinate, all three axes being mutually perpendicular to each other. The line drawn from origin to the point represents the position vector of that point .

Position Vector

In the figure above, the position of a point P is specified and vector OP is called the position vector.

Displacement

Consider a case in which the position of an object changes with time. suppose at certain instant 't' the position of an object is x 1  along the x axis and some other instant 't' the position is x 1  then the displacement δx is defined as,.

Δx = x 2  - x 1

It can be seen in the figure above where x 1  and x 2  are instantaneous position of the object at that time.

Difference Between Distance and Displacement

To understand the difference between distance and displacement, we study the motion of vertical throw of a ball with respect to point O, as shown in the left figure, to height h.

After some time it will come again to the same point O. The displacement of ball is zero but there is some distance traversed by the ball. It's because distance is a scalar quantity but displacement is a vector quantity.

Uniform and Non Uniform Motion

Speed is the rate of change of distance without regard to directions. Velocity is the rate at which the position vector of a particle changes with time. Velocity is a vector quantity whereas speed is scalar quantity but both are measured in the same unit m/sec.

The motion of an object may be uniform or non-uniform depending upon its speed. In case of uniform motion the speed is constant, whereas in the non-uniform motion, the speed is variable.

In uniform motion in one dimension the velocity (v) is mathematically defined as

v = (x 2  - x 1 )/(T-t)     ...... (1)

Where x 1  and x 2  are instantaneous displacement as shown in figure above at time 't' and 'T' respectively.

Graphical Representation of the Uniform Motion

Form the equation (1) we have the following equation.

x 2  = x 1  + v(T - t)

where v is constant. Take t = 0, the equation becomes x 2  = x 1  + vT, from this equation it follows that the graph of position of object 'x 2 ' against 'T' is a straight line, cutting off x 1  on the position axis where x 1  is the distance of the particle from the origin at time t = 0.

v = slope of the graph which is constant

Velocity Vector in Non Uniform Motion 

In any non-uniform motion, we can define an average velocity over a time interval. average velocity   is the ratio of the displacement δx (that occurs during a particle time interval δt) to that interval of time i.e..

Note:   The ratio of total distance traveled and time taken during the motion is called average speed. Average speed is a scalar quantity.

Instantaneous velocity

Instantaneous velocity is defined as the rate of change of displacement.

The velocity at any instant is obtained from the average velocity shrinking the time interval closer to zero. As Δt tends to zero, the average velocity approaches a limiting value, which is the velocity at that instant, called instantaneous velocity, which is a vector quantity, mathematically we can define it as

In the example related with figure given below, the instantaneous velocity is

Speed is defined as rate of change of distance with time.

In any interval of time, average speed is defined as

Think :   (i)Can a body have a constant speed and still have a varying velocity?               (ii)Can a body have a constant velocity and still have a varying speed?

Problem 1 :-

This question contains statement-1 (Assertion) and Statement-2 (Reason). Question has 4 choices (A), (B), (C) and (D) out of which only one is correct.

Statement-1

A bus moving due north take a turn and starts moving towards east with same speed. There will be no change in the velocity of the bus.

Statement-2

Velocity is a vector quantity.

(A)  Statement-1 is true, Statement-2 is true, Statement-2 is a correct explanation for statement-1.

(B)  Statement-1 is true, Statement-2 is true, Statement-2 is not a correct explanation for statement-1.

(C)  Statement-1 is true, Statement-2 is false.

(D)  Statement-1 is false, Statement-2 is true.

Solution:- (D)

This is so because bus is changing its direction of motion.

______________________________________________________________________________________________

Problem 2 :-

A man started running form origin and went up to (2, 0) and returned back to (-1, 0) as shown in figure 2.7. In this process total time taken by man is 2 seconds. Find the average velocity and average speed.

The man is displaced form origin to (-1, 0)

where as, since the total distance traveled by man

= (0, 0) to (2, 0)+(2, 0) to (0, 0)+(0, 0) to (-1, 0)

= 2 + 2 + 1 = 5 m

Hence average speed

= (Total distance)/(Total time)=5/2 m/sec.

Probem 3 :-

A cyclist moves 12 km due to north and then 5 km due east in 3 hr. Find (a) his average speed, (b) average velocity, in m/s.

  In the figure, A shows the initial position and C the final position of the cyclist. The total distance covered by the cyclist AB+BC= (12+5)km = 17 km.

 So, its average speed = 17/3 km/hr = 1.57 m/s

Its displacement is AC and the magnitude is given by

AC = √(AB 2  + BC 2 ) = √(12 2  + 5 2 ) km = 13 km

Thus, its average velocity = 13/3 km/hr

 = 1.2 m/s along AC, i.e at tan -1 (5/12) or 22.6 o  East of North.

_______________________________________________________________________________________________

Problem 4:-

A train is moving with a constant speed of 5 m/s and there are two persons A and B standing at a separation of 10 m inside the train. Another person C is standing on the ground. Then, find

(a)    displacement covered by A, if he moves towards B and back to its position in 10 seconds in frame of reference of train and in frame of reference of C.

 (b)    distance covered by A in frame of reference of train and in frame of reference of C.

(a)    In the frame of train, displacement covered by A is zero and in frame of reference of C, displacement covered by A=0 + 5x10 = 50 m.  

(b)    Distance covered by A in frame of reference of train is 20m and distance covered by A in frame of reference of C is (20 + 50) = 70 m.  

Acceleration

Acceleration is the rate of change of velocity with time. The concept of acceleration is understood in non-uniform motion. It is a vector quantity.

Average acceleration is the change in velocity per unit time over an interval of time.

Instantaneous acceleration is defined as

Acceleration Vector in Non Uniform Motion

Variable Acceleration

The acceleration at any instant is obtained from the average acceleration by shrinking the time interval closer zero. As Δt tends to zero average acceleration approaching a limiting value, which is the acceleration at that instant called instantaneous acceleration which is vector quantity.

i.e. the instantaneous acceleration is the derivative of velocity.

Hence instantaneous acceleration of a particle at any instant is the rate at which its velocity is changing at that instant. Instantaneous acceleration at any point is the slope of the curve v (t) at that point as shown in figure above.

Equations of Motion

The relationship among different parameter like displacement velocity, acceleration can be derived using the concept of average acceleration and concept of average acceleration and instantaneous acceleration.

When acceleration is constant, a distinction between average acceleration and instantaneous acceleration loses its meaning, so we can write

This is the first useful equation of motion.

Similarly for displacement

This is the second important equation of motion.

Now from equation (2), square both side of this equation we get

This is another important equation of motion.

Refer this video for better understanding about motion in one dimension:-

Problem 5 :-

The nucleus of helium atom (alpha-particle) travels inside a straight hollow tube of length 2.0 meters long which forms part of a particle accelerator. (a) If one assumes uniform acceleration, how long is the particle in the tube if it enters at a speed of 1000 meter/sec and leaves at 9000 meter/sec? (b) What is its acceleration during this interval?

(a) We choose x-axis parallel to the tube, its positive direction being that in which the particle is moving and its origin at the tube entrance. We are given x and v x  and we seek t. The acceleration ax is not involved. Hence we use equation 3, x = x 0  + <v> t.

x = v 0  + ½ (v x 0 ) + v x ) t, with x 0  = 0 or

t = 2x/(v x0 +v x) ,

t = ((2)( 2.0 meters))/((1000+9000)meters/sec) = 4.0/10 -4  sec    Ans.

(b)  The acceleration follows from equation 2, v x  = v x 0  + a x t

=> ax = (v 0 -v x0 )/t = ((9000-1000)meters/sec)/(4.0×10 (-4)  sec)

= 2.0 × 10 7  meter/sec 2  Ans.

The above equations of motion are, however, universal and can be derived by using differential calculus as given below:

Thus, we have derived the same equation of motion using calculus.

To understand the use of calculus in solving the kinematics problems we can look into the following illustrations.

__________________________________________________________________________________________________

Problem 6 :-

The displacement x of a particle moving in one dimension, under the action of a constant force is related to the time t by the equation t = √x + 3 where x is in meter and t is in seconds. Find the displacement of the particle when its velocity is zero.

Here t = √x + 3 => √x = t - 3

Squaring both sides, we get x = t 2  - 6t + 9,

As we know velocity, v = dx/dt

Hence we get v = dx/dt = 2t - 6

Put v = 0, we get, 2t - 6 = 0        

When t = 3s, x = t 2  - 6t + 9 = 9 - 6(3) + 9 = 0

Hence the displacement of the particle is zero when its velocity is zero.

_________________________________________________________________________________________________

Problem 7 :-

A particle starts from a point whose initial velocity is v 1  and it reaches with final velocity v 2 , at point B which is at a distance 'd' from point A. The path is straight line. If acceleration is proportional to velocity, find the time taken by particle from A to B.

Here acceleration a is proportional to velocity v.

Hence a α v

=> a = kv, where k is constant

=> dv/dt = kv ............... (1)

=> (dv/ds)(ds/dt) = kv => (dv/ds) v = kv

=> k = (v 2 -v 1 )/d

From equation (1)

=> (dv/v)= k(dt)

or, ln (v 2 /v 1 ) = kt

So, t =  [ln (v 2 /v 1 )]/k = [d/ (v 2 -v 1 )] [ln (v 2 /v 1 )]

Question 1 :-

What determines the nature of path following by the particle?

(a) speed         (b) velocity       (c) acceleration     (d) none of these

Querstion 2 :-

A moving body is covering distances in proportion to the square of time along a straight line. The acceleration of the body is:

(a) increasing    (b) decreasing   (c) zero     (d) constant

Querstion 3 :-

The distance covered by a body in time ‘t’ is propertional to the square of the time ‘t’. The acceleration of the body is:

(a) zero     (b) constant        (c) increasing       (d) decreasing

Querstion 4 :-

A train travels 4 km due east and then 3 km due north and finally comes back to the starting position travelling 5 km along south-west direction. What is the net displacement?

(a) 12 km    (b) 5 km   (c) zero    (d) 1 km

Querstion 5 :-

A body moves along a straight line, the motion of the body is said to be:

(a) one dimensional              (b) two dimensional

(c) three dimensional           (d) may be any of these three.

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