Notes

Kinematics:

          Kinematics is the study of the motion of an object without discussing the cause of motion.

Difference between rest and motion:

Rest:

          A body is said to be at rest if it does not change its position concerning its surroundings.

Motion:

          A body is said to be in motion if it changes its position concerning its surroundings.

The state of rest or motion of a body is relative. For example, a passenger sitting in a moving bus is at rest because he/she is not changing his/her position concerning other passengers or objects on the bus. But to an observer outside the bus, the passengers and the objects inside the bus are in motion.

Surroundings:

            Surroundings are the places in its neighborhood where various objects are present. Similarly

Types of motion:

        There are three types of motion.

1. Translatory motion (linear, random, and circular)
2. Rotatory motion
3. Vibratory motion (to and fro motion)

Translatory motion:

            In translational motion, a body moves along a line without any rotation. The line may be straight or curved.

Examples:

Riders moving in a Ferris wheel are also in translational motion. Their motion is in a circle without rotation.

Types of translator motion:

Translator motions can be divided into linear motion, circular motion, and random motion.

1. Linear motion:

The straight-line motion of a body is known as its linear motion.

Examples:

The motion of objects such as a car moving on a straight and level road is linear motion.

Airplanes flying straight in the air and objects falling vertically down are also examples of linear motion.

2. Circular motion:

The motion of an object in a circular path is known as circular motion

Examples:

A stone tied at the end of a string can be made to whirl. The stone moves in a circle and thus has a circular motion.

A toy train moving on a circular track. A bicycle or a car moving along a circular track possesses circular motion. The motion of the Earth around the Sun and the motion of the moon around the Earth are also examples of circular motions.

3. Random motion:

The disordered or irregular motion of an object is called random motion.

Examples:

The motion of insects and birds is irregular. Thus, the motion of insects and birds is random.

The motion of dust or smoke particles in the air is also random.

The Brownian motion of a gas or liquid molecules along a zig-zag path is also an example of random motion.

Rotatory motion:

            The spinning motion of a body about its axis is called its rotatory motion.

Examples:

The top spins about its axis passing through it and thus it possesses rotatory motion. An axis is a line around which a body rotates. In a circular motion, the point about which body moves about which a body goes around is outside the body. In rotatory motion, the line, around which a body moves about, is passing through the body itself.

The motion of a wheel about its axis and that of a steering wheel are examples of rotatory motion. The motion of the Earth around the Sun is circular and not the spinning motion. However, the motion of Earth about its geographic axis that causes day and night is rotatory.

Differences in circular and rotatory motion:

            Any turning as if on an axis is rotatory motion.  Any rotatory motion where the radius of gyration length and the axis of rotation are fixed is circular motion. And that’s the difference. Circular motion is just a special case of rotatory motion. That is, there is no fixed axis and radius restriction for rotatory motion, but there is for circular motion.

For example, all planets have rotatory motion around their suns. But most of the orbits are elliptical, so the rotation axis (there are two in an ellipse) and radii of gyration vary as they trek around. So most, if not all, planets do not have circular motion.

Note:

Gyration length:

            A length that represents the distance in a rotating system between the point about which it is rotating and the point to or from which a transfer of energy has the maximum effect.

Mini Exercise

       1. When a body is said to be at rest?

Ans: A body is said to be at rest if it does not change its position concerning its surroundings.

       2. Give an example of a body that is at rest and is in motion at the same time.

Ans: Motion and rest are relative concepts. There is no absolute rest. We can define the state of rest or motion only concerning another object or a point in space taken as reference.

Examples:

• A person inside a train considers himself to be at rest concerning the fellow passengers or the walls of the train. But when he looks outside, he finds himself to be in motion concerning the trees outside.
• A passenger sitting in a moving bus is at rest because he/she is not changing his/her position concerning other passengers or objects inside the bus. But to an observer outside the bus, the passengers and the objects inside the bus are in motion.

      3. Mention the type of motion in each of the following:

• A ball moving vertically upward.

Ans: Linear motion.

• A child moving down a slide.

Ans: Linear motion.

• Movement of a player on a football ground.

Ans: Random motion.

• The flight of a butterfly.

Ans: Random motion.

• An athlete running on a circular track.

Ans: Circular motion.

• The motion of a wheel.

Ans: Circular motion.

• The motion of a cradle.

Ans: Vibratory motion.

Vibratory motion:

            To and fro motion of a body about its mean position is known as vibratory motion.

Examples:

Consider the motion of a baby about in a swing. As it is pushed, the swing moves back and forth about its mean position. The motion of the baby repeats from one extreme to the other extreme with the swing. Such a type of motion is called vibratory motion.

To and fro the motion of the pendulum of a clock about its mean position, is called vibratory motion.

A baby in a cradle moving to and fro, to and fro motion of the hammer of a ringing electric bell and the motion of the string of a sitar are some of the examples of vibratory motion.   

Differentiate between scalars and vectors

Representation of vectors (Symbolic representation of a vector):

To differentiate a vector from a scalar quantity we generally use bold letters to represent vector quantities, such as F, a, d, or a bar or arrow over their symbols such as

$\bar{F},\bar{a},\bar{d}$ or $\overrightarrow{a}$ and $\overrightarrow{d}$

Vector representation/Graphical representation of a vector:

A straight line is drawn with an arrowhead at one end. According to some suitable scale, the length of the line represents the magnitude and the arrowhead gives the direction of the vector.

Position:

            The term position describes the location of a place or a point concerning some reference point called the origin.

For example:

You want to describe the position of your school from your home. Let the school be represented by S and home by H. The position of your school from your home will be represented by a straight-line HS in the direction from H to S.

Difference between distance and displacement:

Uniform speed:

            A body has uniform speed if it covers equal distances in equal intervals of time however short the interval may be.

variable speed:

          If a body covers unequal distances in an equal interval of time, however, small the interval may be, the speed of the body is said to be variable.

Average speed:

            The ratio between distance and total time taken is known as average speed.

    Average speed=  $\frac{Total\: distance \:coverd}{Total \: time \: taken}$

      Vay=  $\frac{S}{T}$

Uniform velocity:

            A body has uniform velocity if it covers equal displacement in equal intervals of time however short the interval may be.

Variable velocity:

            If speed or direction changes with time then the velocity of such a body is said to be variable.

Average velocity:

The ratio between displacement and time is known as the average velocity

Average speed= $\frac{Distance}{Time}$

Vav= $\frac{D}{T}$

Acceleration:

Acceleration is defined as the rate of change of velocity of a body.

Acceleration= $\frac{Change \: in \; Velocity}{Time \; Taken}$

Acceleration= $\frac{Final Velocity \: - \; Initial Velocity}{Time \: Taken}$

             a= $\frac{Vf \: - \: Vi }{T}$

Unit of acceleration:

          SI unit of acceleration is metre per second squared (ms^-2)

USEFUL INFORMATION

Acceleration of a moving object is in the direction of the velocity of its velocity is increasing. Acceleration of the object is opposite to the direction of velocity if its velocity is decreasing.

Uniform acceleration:

A body has uniform acceleration if it has equal changes in velocity in equal intervals of time however short the interval may be.

Positive acceleration:

The acceleration of a body is positive if its velocity increases with time. The direction of this acceleration is the same in which the body is moving without a change in its direction.

Negative acceleration/Deceleration or retardation:

Acceleration of a body is negative if the velocity of the body decreases. The direction of negative acceleration is opposite to the direction in which the body is moving. Negative acceleration is also called deceleration or retardation.

DO YOU KNOW

A graph may also be used in everyday life such as to show year-wise growth/decline of export, month-wise rainfall, a patient’s temperature record or runs per over scored by a team, and so on.

Graph:

The graph is a pictorial way of presenting information about the relationship between various quantities.

Variables:

The quantities between which a graph is plotted are called the variables.

Independent quantity:

One of the quantities is called independent quantity.

Dependent quantity:

The values which vary with the independent quantity are called dependent quantity.

Distance-time graph:

            It is useful to represent the motion of objects using graphs. The terms distance and displacement are used interchangeably when the motion is in a straight line. Similarly, if the motion is in a straight line then speed and velocity are also used interchangeably

Note:

In a distance-time graph, time is taken along the horizontal axis while the vertical axis shows the distance covered by the object.

Object at rest:

            In the graph shown in the figure, the distance moved by the object with time is zero. That is, the object is at rest. Thus, a horizontal line parallel to the time axis on a distance-time graph shows the speed of the object is zero.

An object moving with constant speed:

The speed of an object is said to be constant if it covers equal distances in equal intervals of time. The distance-time graph as shown in the figure is a straight line.

Its slope gives the speed of the object.

Consider two points A and B on the graph.

Speed of the objects = slope of line AB=  $\frac{Distance \: EF}{Time\: CD}  =\frac{20m}{10s} =2 \frac{2m}{s}$

The speed found in the graph is 2 ms^-1

An object moving with variable speed:

When an object does not cover equal distances in equal intervals of time then its speed is not constant. In this case, the distance-time graph is not a straight line as shown in the figure. The slope of the curve at any point can be found from the slope of the tangent at that point. For example.

The slope of the tangent at P= $\frac{RS}{QS} = \frac{30m}{10s} = 3m/s$

Thus, the speed of the object at P is 3 ms^-1

Note:

The speed is higher at instants when the slope is greater, speed is zero at instants when the slope is horizontal.

Speed-time graph:

          In a speed-time graph, time is taken along the x-axis, and speed is taken along the y-axis.

Object moving with constant speed

When the speed of an object is constant (4ms^-1) with time, then the speed-time graph will be a horizontal line parallel to the time-axis along the x-axis.

A straight line parallel to the time-axis represents the constant speed of the object.

An object moving with uniformly changing speed (uniform acceleration):

Let the speed of an object be changing uniformly. In such a case speed is changing at a constant rate. Thus, its speed-time graph would be a straight line.

A straight line means that the object is moving with uniform acceleration. The slope of the line gives the magnitude of its acceleration.

Speed-time graph gives a negative slope. Thus, the object has a deceleration of 0.4ms^-2.

Distance traveled by a moving object:

            The area under a speed-time graph represents the distance traveled by the object. If the motion is uniform then the area can be calculated using the appropriate formula for geometrical shapes represented by the graph.

Equations of motion:

          There are three basic equations of motion for bodies moving with uniform acceleration. These equations relate to initial velocity, final velocity, acceleration, time, and distance covered by a moving body.

Suppose a body is moving with initial velocity v_i, and after time t its velocity becomes v_f. Then acceleration a is given by

a= $\frac{VF \: VI}{T}$

Or  Vf - Vi = at 

Vf = Vi + at

Second Method (Graphical method):

The first equation of motion:

 Speed-time graph for the motion of a body is shown in the figure. The slope of line AB gives the acceleration of a body.

Slope of line AB= a=  $\frac{AB}{AC}$ = $\frac{BD \: - \: CD}{OD}$

As

BD = V_f,      CD = V_i and     OD = t

Hence, a= $\frac{VF \: VI}{T}$

Or, V_f - V_i = at

V_f = V_i + at

Suppose a body is moving with initial velocity v_i and after a certain time t its velocity becomes v_f then the total distance S covered in time t is given by

$S = v_{av} \times t$

\begin{equation} S = \dfrac{vf + Vi}{2} \times t \end{equation}

From the first equation of motion.                 V_f = V_i + at   

Putting the value of V_f in equation (1).

$\dfrac{vf + at vi}{2} \times t$

$\dfrac{2vi+ at}{2} \times t$

$\dfrac{2vit+ at2}{2} \times t$

$\dfrac{2vit}{2}$ + $\dfrac{at2}{2}$

S = V.t + ½ at²

Second Method (Graphical method):

The second equation of motion:

In the speed-time graph shown in the figure, the total distance S travelled by the body is equal to the total area OABD under the graph. That is

Total distance S = area of (rectangle OACD + triangle ABC)

Area of rectangle OACD = OA x OD

                                           = V_i x t

Area of triangle ABC       = ½ (AC x BC)

                                           = ½ t x at

Since Total area OABD = area of rectangle OACD + area of triangle ABC

Putting values in the above equation, we get

                          S = V_it + ½ t x at

                          S = V_it + ½ at²

Suppose a body is moving with initial velocity v_i and after a certain time t its velocity becomes v_f then the total distance S covered in time t is given by

S = ${vav}\times t$

S = $\frac{vf \: + \: vi}{2}\times t$ ……..(1)

From the first equation of motion find the value of t.      

vf = vi +at      Or     t = $S=\frac{vf \: - \: vi}{a}$

Putting the value of V_f in equation (1).

S = $\frac{vf \: + \: vi}{2}\times\frac{vf \: - \: vi}{a}$

2as = $(vf+vi)\times (vf - vi)$   by using formula   2as = $(a+b)(a-b)=a^{2}-b^{2}$

2as = $vf^{2}-vi^{2}$

Second Method (Graphical method)

Third equation of motion:

In the speed-time graph shown in the figure, the total distance S traveled by the body is given by the total area OABD under the graph.

Total area $OABD = S=\left(\frac{OA + BD}{2} \right)\times OD$

Or          $2S = (OA + BD) x OD$

Multiply both sides by BC/OD, we get:               

$\frac{BC}{OD} =a$

$2S\times\frac{BC}{OD} =\left(OA + BD\right)\times OD\times\frac{BC}{OD}$

$2S\times\frac{BC}{OD} =\left(OA + BD\right)\times BC$ .........(1)

Putting the value in the above equation (1), we get

$2S\times a =\left(Vi + Vf\right)\times \left(Vf - Vi\right)$

$2aS = Vf^{2} + Vi^{2}$

USEFUL INFORMATION

• To convert ms^-1 to kmh^-1

1 ms^-1 = 0.001km x 3600 = 3.6 kmh^-1

Thus, multiply speed in ms^-1 by 3.6 to get speed in km^-1 e.g.,

20 ms^-1= 20 x 3.6 kmh^-1=72 kmh^-1

• To convert kmh^-1 to ms^-1

^{$1 kmh^{-1}= \frac{1000m}{60x60s} =\frac{10}{36}ms^{-1}$}

Thus, multiply speed in kmh^-1 by  to get speed in ms^-1 e.g.

$50 kmh^{-1}= 50 \times \frac{10}{36} ms^{-1}= 13.88 ms^{-1}$

• To convert ms^-2 to kmh^-2

Multiply acceleration in ms^-2 by $\frac{3600\times3600}{1000} = 12960$ to get its value in kmh^-2

• To convert km^-2 to ms^-2

Divide acceleration in kmh^-2 by 12960 to get its value in ms^-2

The velocity of an object will increase due to the earth's gravity. That is why bodies falling freely g is positive.

The motion of freely falling bodies:

          The acceleration of freely falling bodies is called gravitational acceleration.

It is denoted by g. On the surface of the Earth, its value is approximately 10 ms^-2.

            For bodies falling freely, g is positive and is negative for bodies moving up.

            Galileo was the first scientist to notice that all the freely falling objects have the same acceleration independent of their masses. He dropped various objects of different masses from the leaning tower of Pisa. He found that all of them reach the ground at the same time.

Equations of motion for bodies moving under gravity:

1. V_f = V_i + gt
2. h = V_it + ½ gt²
3. 2gh = V_f² = V_i²