Essence of Physics, Chemistry, and Mathematics

By Dr. Akhilesh Sharma, Yogendra Mohan and S.B. Singh

This book has three sections namely Physics, Chemistry, and Mathematics having 15, 6, and 13 chapters respectively with illustrations.

The book contains the p

• Language: English
• Publisher: Blue Rose Publishers
• Release date: Apr 24, 2024
• ISBN: 9789362615503

Book preview

Chapter 1: Measurement in Science and Technology

1.1 Measurement:

Measurement is the comparison one of unknown physical quantity with a known fixed unit quantity. For example, if the length of a rod is 3 meter it means that the unit of the length is meter and this unit is contained three times in the length of the rod.

Hence, to express the magnitude of a physical quantity, we should know two things:

The unit in which the quantity is measured.

The numerical value (which express that how many times the above mentioned units is contained in the given physical quantity).

Therefore, Magnitude of a physical quantity,(Q)=Numerical value of physical quntity(n)*size of its unit(u).

i.e. Q=nu

1.2 Unit:

The quantity used as a standard of measurement is called the unit.

1.3 Fundamental units:

The units selected for measuring mass, length and time are called fundamental units.

These three units are independent of each other and are not definable in terms of other quantities. So, these units are called fundamental or base units.

There are seven basic or fundamental units for seven basic quantities. The seven basic quantities are:

Mass

Time

Temperature

Current

Amount of substance, and

Luminous intensity

1.3.1 Derived unit: The units of physical quantities which can be expressed in terms of fundamental units are called derived unit. For example

1.3.2 Characteristics of a standard unit:

It should be of convenient size.

It should be well defined.

It should be easily available.

It should not change with time and place.

It should not change with change in physical conditions (e.g. temperature, pressure, humidity, mechanical stress etc.).

It should be universally agreed upon so that results obtained in different countries are comparable.

1.3.3 Systems or fundamental units or measurement systems: There are three systems. They are:

CGS system (Centimeter for length, gram for mass and second for time).

MKS system (Meter for length, kilogram for mass and second for time).

FPS system (Foot for Length, Pound for mass and second for time).

1.3.4 SI system (System International) units

These are introduced in 1960 and are acceptable by all countries for all kinds of scientific works.

1.3.5 Base units: The base units are shown in table 1.1.

Table 1.1: Base unit

1.3.6 Supplementary units:

Plane angle measured in radian (rad/Rad)

Solid angle measured in steradian (Sr)

1.4 Unit of length

1.4.1 Meter: In 1889, a meter was originally defined as the distance between two marks drawn on a Platinum-Iridium (an alloy with 90% Platinum and 10% Iridium rod kept at zero degrees centigrade in the International Bureau of Weights and Measures at Serves near Paris.

In 1960, the meter was redefined as 1650763.73 times the wavelength of specified orange-red spectral line in the emission spectrum of Krypton-86 or 1 meter is 1553164.1 times the wavelength of red line in spectrum of Cadmium.

In 1983, the meter has been redefined in the terms of speed of light, according to which the distance which the light travels in 1/299792458 of a second in air (or vacuum) is called one meter.

1 centimetre (Cm) = 10-2m

1 Millimetre (mm) = 10-3m

1 Micro (µ) = 10-6m

1 Nanometre (nm) = 10-9m

1 Angstrom (A⁰) = 10-10m

1 Fermi (f) = 10-15m

1 Astronomical unit (A.U.) =1.496¹¹m

1 light year (ly) = 9.46¹⁵m

1 Parsec = 3.26 * light year = 3.24 * 9.46 * 10¹⁵ m = 3.08 * 10¹⁶m

1.4.2 Unit of mass

1.4.3 Kilogram: In 1889, one kilogram was defined as the mass of cylindrical piece of Platinum- Iridium alloy kept in International Bureau of Weights and Measures at Sevres near Paris. However, the mass of 1 liter of water at 4 degrees centigrade is also taken as one kilogram.

1 gram (gm) = 10-3 kg

1 milligram (mg)=10-6 kg

1 quintal=100 kg

1 metric tonne (MT)=1000 kg

One atomic mass unit is defined as (1/12)th the mass of 1 Carbon -12 atom.

1 atomic mass unit (a.m.u. or u)=1.655*10-27 kg

1 solar mass =2.0 * 10³⁰ kg

1.4.4 Unit of time: Second: A second is defined as (1/86400)th part of a mean solar day.

1.5 Derived unit:

These are obtained by the combination of one, two or more fundamental units. Some derived units are named after the names of the scientists who invented them, for example, Newton for force (kg-ms-2), Joule for energy (kg-m²s-2), etc.

Chapter 2: Description of Atomic Structure

2.1 Fundamental Experiments and discharge tube and the discovery of electron

William Crookes noted that gases are ordinarily poor conductors of electricity. However, when a high voltage (10000 Volt) charge from an induction coil is applied to tubes filled with gases it very low pressure (0.01 mm of mercury), the gas becomes good conductors of electricity and begin to flow in the form of rays. Since these rays originate from the negative plate (i.e. cathode) and travels towards the anode, they are called Cathode Ray.

J J Thompson he studied the properties and constituents of cathode ray. He used a tube, called as discharge tube or a Cathode ray tube. This is as shown in Fig. 2.1.

Fig. 2.1: Discharge tube in which electrons are flowing

2.2 Properties of Cathode Rays

They travel from cathode to the anode in a straight line next property.

They are affected by electric field i.e. they are attracted toward positive field and deflected from negative field. This shows that they carry negative charges.

They cause a greenish yellow fluorescence on a soda-glass screen, placed in the tube

They exert mechanical pressure and transfer sufficient kinetic energy to heat up the metallic object on which they fall.

They produce X-ray when they fall upon hard metallic targets like tungsten.

They penetrate through matter.

They cause ionization of the gas through which they pass.

The ratio off the charge (e) to mass of the particles constituting cathode ray remains the same irrespective of the nature of the gas taken and of the metal forming the cathode.

2.2.1 Thompson concluded that

Cathode rays consist of negatively charged particle now called electrons.

These negatively charged particles are integral part of all items.

Electrons have both definite mass and definite electric charge, both of which are independent of the nature of the gas in the discharge tube.

2.2.2 Properties of electrons

Electrons from all sources are alike, having identical mass.

They are constituent part of all atoms.

The mass of an electron is 1/1837 the mass of a hydrogen atom (or 9.108*10 -31 ).

An electron carries unit negative charge of magnitude -1.602*10 -19 coulombs.

The electron is extremely small; its radius is less than 1*10 -15 meter.

An electron is represented by e or e - .

2.3 Canal rays or positive rays or anode rays and discovery of protons

In 1886, Goldstein noticed that when a perforated cathode ray was used in the experiment of discharge tube, as shown in fig. 2.2, another set of rays traveling in a direction opposite to that of the cathode ray i.e. from anode towards the cathode was seen. He called these rays as canal rays. Since these passed through holes on canals in the cathode. Later on, these rays were named as positive ray or an anode ray.

Fig. 2.2: Construction of Canal Ray

2.3.1 Properties of canal or anode or positive rays

Anode rays travel in a straight line.

They consist of minute material particles and hence produce mechanical effect.

They are made-up of positively charged particles.

Positive rays are reflected by electric and magnetic fields but in a direction opposite to that of the cathode rays. This means that these rays consist of positively charged particle called proton.

These rays produce fluorescence on a zinc sulphide screen.

There e/m i.e. charge to mass ratio differs from gas to gas. Its value is much less than that of and electron and is maximum when hydrogen is taken in the discharge tube.

2.3.2 Properties of proton

It possesses a unit positive charge of the value 1.602*10 -19 coulombs.

Its mass is same as that of a hydrogen atom 1 amu ( A tomic M ass U nit) i.e. 1837 times the mass of an electron which is 1.672*10 -27 Kg.

The proton resides in the central part of an atom i.e. in the nucleus.

2.4 Discovery of neutrons

We know that an atom contains electrons and protons and that the atomic mass of an electron is negligible. Therefore, an atom of helium which contains 2 protons should have a mass equals to 2*1 amu = 2 amu. But the atomic mass of a helium atom was found to be approximately 4 amu. It was therefore proved that in the nucleus of an atom there is another particle called as neutron which possesses no electrical charge but is almost of an equal in mass as the proton.

In 1932, Chadwick discovered these neutral particles by bombarding light nuclei like beryllium with alpha particles i.e. helium nuclei.

2.4.1 Properties of neutrons

These particles were not found to be deflected by any magnetic or electric field proving that it is electrically neutral.

The mass of a neutron is equal to 1.676*10 -27 .

2.4.2 Structure of an atom

The atom is built out of a number of sub atomic particles. The three sub atomic particles of great importance in understanding the structure of an atom are electrons (e), protons (p) and neutrons (n). The properties of these subtopic particles are as tabulated in table 2.1.

Table 2.1: Sub atomic particles with their characteristics

2.5 Modern atomic theory

Atoms are divisible into sub-atomic particles like protons, electrons and neutrons.

Atoms can be created and destroyed by nuclear fusion and fission.

The atoms of an element may not be alike in all respects, as is seen in the case of isotopes.

In the formation of organic compounds, the laws of chemical combination are not always followed.

2.5.1 Atomic nucleus and orbit

There are two structural parts of an atom: nucleus and its empty space in which there are imaginary paths, called orbits. The protons and neutrons [collectively known as nucleons] are found in the central part which is known as nucleus of the atom. The orbits are the imaginary paths where electrons revolve around the nucleus of the atom.

2.5.2 Atomic number [Z]

Atomic number of an element is the number of

Protons present in the nucleus of its atom.

Electrons present in its atoms

Positive charges in the nucleus of its atom

2.5.3 Mass number or Atomic mass number [A]

It is the total number of protons and neutrons present in its atomic nucleus.

The mass number however is a whole number approximation of the atomic mass calculated in atomic mass units

2.6 Symbolic representation of an element

An element is characterized by its atomic mass, atomic number and its symbol. This is represented as:

Where, X denotes element

Superscript ‘A’ denotes mass number and

Subscript ‘Z’ denotes atomic number

Atomic Number [Z] = number of protons [P] = number of electrons [e]

i.e, Z = P = e

Atomic mass number [A] = number of protons [P] + number of neutrons [N]

A = P + N = Z + N

2.7 Main features of the structure of atoms

Atoms of all elements (except hydrogen) are made-up of three fundamental (sub-atomic) particles: electron, proton and neutrons.

The nucleus is located at the center of the atom. It contains protons and neutrons which account for the total mass of that atom.

The nucleus is positively charged due to the presence of proton in it.

The electrons are outside the nucleus and have negligible mass.

The number of electrons in an atom is equal to the number of protons in it. Hence the atom is electrically neutral.

The electrons revolve rapidly around the nucleus in fixed circular paths, called energy level or shells or orbits.

The atoms of different elements contain different number of electrons, protons and neutrons.

2.8 Isotopes

An element showing similar chemical properties but different masses is said to be show isotopic and the varieties of the atom are called isotopes.

The isotopes are the atoms of the same element having the same atomic number but different mass number.

Isotopes differ only in their number of neutrons. Some examples are tabulated in table2.2.

Table 2.2: Some isotopes

2.8.1 Important: The isotopes of an element have the same chemical properties but their physical properties are different.

2.8.2 Uses of Isotopes

Some isotopes are radioactive (isotopes of cobalt) and are used for treating cancer and other diseases.

is used for determining the age of historical and geological material.

An isotope of uranium is used as a fuel in nuclear reactors.

An isotope of iodine is used in the treatment of goiter.

2.8.3 Isobars

They are atoms of different elements with the same mass number but different atomic number and has add such they have different physical and chemical properties for example,

2.9 X-rays

When a stream of fast moving electrons (Cathode rays) fall on a tungsten or platinum target of high atomic weight, X-rays are produced but only 1% of the incident energy of electron is converted into X-rays. Remaining energy appears as heat. X-rays are electromagnetic radiations of very sort wavelengths.

2.9.1 Production of X-rays

Coolidge tube is commonly used to produce X-ray. It consists of an evacuated glass bulb consisting of a cathode ‘C’ and target ‘T’. The cathode is a tungsten filament and can be heated by the current supplied by the low tension DC battery or by a step down transformer. The electrons are emitted from the cathode by the process of thermionic emission. These electrons are highly accelerated by maintaining a high potential difference (about 10,000 volts) between the cathode and the target. These accelerated electrons strike a tungsten target (T). This results in X-rays production, as shown in fig. 2.3.

Fig. 2.3: Generation of X-ray

2.9.2 Properties of X-rays

They are electromagnetic radiations of very small wavelengths of the order 10 -10 meter and are protons of high energy.

They travel with the velocity of light (3*10 ⁸ meter per second).

They affect the photographic plate.

They are not deflected by magnetic or electric field and therefore do not opposes any charge.

Similar to light X-rays, due to their energy, liberate photoelectrons from some metals when allowed to fall on them.

They ionized the gas through which they pass.

They produce fluorescence in barium Platinocynide, Zinc sulphide and cadmium tungstate.

They are highly penetrating and can pass through many solids which are opaque to visible light viz wood, flesh, paper, cardboard, thin sheets of metals, ebonite etc.

They show interference, diffraction and polarization, similar to light radiations.

They have destructive effect on living tissues and long exposure of the skin to X-rays is harmful. In some cases, X-rays produce reddening of the scale and kill white corpuscles of the blood.

2.9.3 Uses of X-ray

Surgery: X-rays can pass through blood and not through bones. They are used to detect the fracture of bones, diseases, organic and foreign bodies and growth in the human bodies.

Radiotherapy: X-rays are used to destroy malignant tumors and to cure skin diseases. Long exposure to X-rays kills the germs in the body and hard X-rays are used to destroy tumors very deep inside the body.

Industry: X-rays are used to detect any defects in the radio valves, tennis balls, rubber tires and the presence of pearls in oysters.

Engineering: X-rays are used to detect cracks in structures and blow holes in metals. They are used to test the quality of welding, molds and metal castings.

Detective departments: X-rays are commonly used to detect the smuggling of precious metals at the custom posts and to detect the explosive and other contrabands goods like opium in sealed parcels and in leather cases.

Mints: X-rays are used in mints where coins are made and every person has to pass before an X-ray unit after the day’s work is over.

Research: X-rays are used in research to study the structure of crystals arrangement of atoms and molecules in matter and their behavior on different materials.

2.10 Radioactivity

The phenomena of spontaneous disintegration of the nucleus of an atom with the emission of some radiation are called radio activity. The substances having this property are called radioactive substances. Uranium, thorium, radium, etc. are radioactive substances.

During radiation (radioactivity), alpha (α-ray), beta (β-ray) and gamma rays (γ-rays) are produced.

2.10.1 Properties of alpha particles

They have positive charge of +2 units.

They travel with a velocity of 3*10 ⁷ meter per second i.e. 1/10 th the speed of light.

Their penetrating power is less. These rays are stopped by aluminum foils of 0.01 millimeter thickness.

These rays possess strong ionizing power i.e. they take out electrons from the gas through which they pass.

These particles darken the photographic plate.

These are nuclei of helium .

2.10.2 Properties of beta particles

These are electrons .

They possess negligible mass

The velocity of beta particle is 3*10 ⁸ m/s i.e. equal to the velocity of light.

Their penetrating power is 100 times more than that of the alpha particles.

Beta particles possess feeble (negligible) ionizing power.

Beta particles affect the photographic films rapidly.

2.10.3 Particles of gamma particles

They are electromagnetic waves .

They are neutral in nature. They do not possess any charge.

They travel with the velocity of light.

They possess very high penetrating power.

They do not ionize the material through which they pass.

Gamma rays darken the photographic plates rapidly.

2.11 Natural radioactivity

The radioactivity exhibited by some naturally occurring element is called natural radioactivity. All elements with atomic number greater than 82 possess natural radioactivity.

2.11.1 Induced or artificial radioactivity

The radioactivity which can be induced in elements having atomic number less than 82 by artificial means is called artificial or induced radioactivity.

Chapter 3: Motion

3.1 Rest

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

3.1.1 Motion

A body is said to be in motion if it changes its position with respect to its immediate surroundings. For a moving body, if the distance traveled in a certain time-interval is much large as compared to the size of the body, the body can be assumed to be a point particle.

3.2 One-dimensional motion

When a body moves along a straight line path, its motion is said to be the one-dimensional motion or motion in a straight light or rectilinear motion.

3.2.1 Representation of 1 dimensional motion

The part of one dimensional motion can be represented by a straight line parallel to the X-axis if X-axis is taken in the direction of motion. For example

Table 3.1: Position and time

In the above example, position of a pebble falling freely vertically downwards at different instant is given.

The motion of the pebble can be represented by choosing a proper scale for X on a straight line along X axis. Here X-axis represents the vertical download direction, as shown in the Fig. 3.1.

Fig. 3.1: Plot of the table 1

3.3 Distance

The length of the path transversed by a body is called the distance traveled by it. The path may not be straight. It is a scalar quantity and it is generally represented by S and its SI unit is meter.

3.3.1 Displacement

The shortest distance from the initial to the final position of the body is called the magnitude of displacement. It is in direction from the initial position to the final position. It is a vector quantity and is represented by the symbol S ⃗ and its SI unit is meter

Here, in the adjacent figure, shown in Fig. 3.2, length of path AB is the distance while dotted line from A to B shows the displacement. Here point A is called the origin and point B is called the terminus (in the sense of displacement).

Fig. 3.2: Path

3.3.2 Important points

The magnitude of displacement is either equal to or less than the distance. For example

Distance from O to A = 4km

Displacement from O to A = 4km (east)

Displacement from O to B = 7 km

Displacement from O to B = 5km (North East). This is shown in Fig. 3.3.

Fig: 3.3: Triangle representation of distance

The displacement can be zero even if the distance is not zero. For example, consider the Fig. 3.4.

Distance from A to B = 3m

Displacement from A to B = 3m (upward)

Distance from A to B and B to A = 3km + 3km = 6m

Displacement from A to B and B to A = 0 m

Fig. 3.4: Distance from a point

3.3.3 Uniform Motion

A body is said to be moving with uniform motion if it covers equal distance in equal intervals of time.

3.3.4 Non-Uniform Motion or Variable Motion

A body is said to be moving with non - uniform motion or variable motion if it covers unequal distance in equal intervals of time.

A uniform and non-uniform motion is shown in Fig. 3.5 (a) and (b) respectively. It is seen that the distance (30 m each) is covered in equal time interval (5 sec), making it a uniform motion while, this is not the case for a non-uniform motion where different distances (40 m, 30m and 50m) are covered in 5 sec each.

Fig. 3.5 (a): Uniform and (b): Non-uniform motions

3.3.5 Difference between distance and displacement: It is listed in table 3.2.

Table 3.2: Difference between distance and displacement

3.4 Speed

The speed of the body is the distance travelled by the body in a unit time interval i.e. it is the distance travelled by the body in one second.

It is a scalar quantity and is measured in m/s (metre/second). It is usually represented by the letter v or u. Mathematically, it is expressed as:

A speed may be uniform or non-uniform.

3.4.1 Uniform speed

A body is said to be moving with uniform speed if it covers equal distance in equal time interval. As seen in Fig. 3.5 (a), the body is moving from A to B, a distance of 30m in 5 sec; B to C (30m distance) in 5 sec and C to D (a distance of 30m) in 5 sec. so its speed (6 m/s) is uniform speed.

For example: Motion of a ball on a frictionless plane surface is with uniform speed. If a body moves with uniform speed v, the distance travelled by it in time t is given by d = v*t.

3.4.2 Non-Uniform speed or Variable speed

A body is said to be moving with non-uniform speed if it covers unequal distance in equal time intervals. As seen in Fig. 3.5 (b), the body is moving from A to B, a distance of 40m in 5 sec; B to C (30m distance) in 5 sec and C to D (a distance of 50m) in 5 sec. so its speed are 8 m/s; 6 m/s and 10 m/s. These speeds are different but the time interval is same hence it is a non-uniform speed.

For example: Motion of a ball on a rough plane surface or motion of a car in a crowed street is example of a non-uniform speed.

In a case of a non-uniform speed one can specify the instantaneous speed and average speed of a body.

3.4.3 Instantaneous speed

If the speed of the body keeps on changing continuously with time, its speed at any instant is known as instantaneous speed. The speedometer of a vehicle measures the instantaneous speed.

3.4.4 Average speed

The ratio of the total distance travelled by the body to the total time of the journey is called the average speed. Mathematically, the average speed, Av is expressed as:

In Fig. 3.5 (b), it is seen that the total distance travelled from A to D is 120m [= (40m+30m+50m)]) and total time is 15 second [= (5 +5+5)] so the average speed is 8 m/s [ = (120/15 = 8 m/s].

In case of a body moving with the uniform speed, the instantaneous speed and average speed is equal.

3.4.5 Velocity

The velocity of the body is the distance travelled by the body in a specified direction in a unit time travel. It is also defined as the rate of change of displacement with time is called the velocity i.e. it is the displacement of the body in 1 second.

Velocity is a vector quantity. It is represented by the symbol . Its unit is same as speed, m/s.

For velocity, both its magnitude and the direction must be known. If two bodies are moving with the same speed but in different directions, then their velocities will be different.

3.4.5.1 Uniform Velocity

If a body travels equal distances in equal intervals of time along a particular direction, the body is said to be moving with uniform velocity.

If a body moves with uniform velocity the displacement of the body in a time interval t is given by

3.4.5.2 Non - Uniform Velocity or Variable velocity

If a body travels unequal distances in a particular direction in equal intervals of time or it moves equal distances in equal interval of time but in different direction, then the velocity of the body is said to be variable or non – uniform velocity.

Example: Motion of a freely falling body (here speed increases but direction of the motion remains the same, then the velocity of the body is said to be variable on non-uniform velocity) and motion of a body in a circular path with uniform speed (here speed is constant but the direction continuously changes).

3.4.5.3 Instantaneous velocity

For a body moving with variable velocity, the velocity of the body at any instant is called the instantaneous velocity.

3.4.5.4 Average velocity

If the velocity of a body, moving in a particular direction, changes with time, the ratio of the displacement to the time taken for the whole journey is called the average velocity. Mathematically, average velocity, Av is expressed as:

3.4.5.5 Difference between speed and velocity: It is tabulated in table 3.3.

Table 3.3: Speed and Velocity

3.5 Acceleration

It is the rate of change of velocity with time i.e. it is change in velocity in one second so its unit is m/s² (Metre per Second Square) and is a vector quantity. Symbolically it is represented by and mathematically, it is expressed as:

If the velocity of a body increases with time the motion is said to be accelerated and one obtains positive acceleration but if the velocity of the body decreases with time than it is said to be deceleration (retardation). It is also termed as negative acceleration.

3.5.1 NB:

The velocity of the body determines its direction of the motion. The acceleration does not determine its direction of motion. The positive and negative sign of acceleration simply means the velocity is either increasing or decreasing with time whereas the positive and negative sign of velocity depends on its direction of motion.

3.5.2 Uniform Acceleration

The acceleration is said to be uniform (or constant) when changes in velocity takes place equal in equal interval of time. Example is the motion of a body under gravity (i.e. free fall of a body) is an example of uniformly accelerated motion.

Graphically it is represented as shown in Fig. 3.6.

Fig. 3.6: Graphical representation of uniformly accelerated motion

3.5.3 Variable Acceleration

If the change in the velocity is not the same in the same intervals of time, then it is said to be variable acceleration. For example, the acceleration of a vehicle in a crowded road is variable hence it is an example of a variable acceleration as shown in Fig. 3.7.

Fig. 3.7: Variable acceleration

3.5.4 Average Acceleration

The average rate of change of velocity with respect to time is called average acceleration. Mathematically, if aav is average acceleration, v1(t), v2(t) are the velocities at t1 and t2 respectively, then

Graphs and their uses:

If a body moves in a straight line, its motion is said to be one dimensional (linear or rectilinear motion). The linear motion of the body can be studied with the help of the following graphs.

(i) Displacement time graph

(ii) Velocity time graph

(iii) Acceleration time graph

It should be noted here that for the motion in the one direction in a straight line, the direction of motion does not change. So displacement - time graph and the distance - time graph are the same. Similarly, the velocity -time graph and the speed – time graph are same.

3.6 Displacement -Time graph

In this case, time is taken on the X-axis and the displacement of the body is taken on Y-axis. From the graph, velocity of the body may be determined which the nothing but the slope of the displacement-time graph. If the slope is positive, the body is said to move away from the reference point (starting point). If the slope is negative it means the body is moving towards the reference point.

Case 1: If the body is stationary and the displacement at any instant is same as that at t = 0, as shown in Fig. 3.8, then the displacement time graph is a straight line parallel to the time axis.

Fig. 3.8: Parallel to the time axis

Case 2: If the body is moving with uniform velocity, its displacement increases by the same amount in each second. This displacement-time graph, in this case, is said to be a straight line inclined to the time axis. The velocity of the body can be obtained by finding the slope of the straight line, as shown in the Fig. 3.9.

Fig. 3.9: Linear displacement

Case 3: If a body moves with a variable velocity, the displacement-time graph is not a straight line, but it is a curve. The velocity at any instant can then be obtained by finding the slope (or the gradient) of the tangent drawn on the curve at that instant, as shown in Fig. 3.10.

Fig. 3.10: Variable displacement

NB: the displacement – time graph can never be a straight line parallel to displacement axis because it would mean that the distance covered by the body in a certain direction is increasing without any increase in time i.e. the velocity of the body is infinite which is impossible.

3.7 Velocity -Time Graph

In this case also, the velocity is drawn on y axis and X-axis represents time. Since velocity is a vector quantity, the positive velocity means that the body is moving in a certain direction away from its initial position and negative velocity means that the body is moving in the opposite direction (i.e. towards the initial position).

From the slope of a velocity-time graph we can determine the displacement or distance travelled.

Case 1: If the body moving with uniform velocity at any instant is same as that at t = 0, as shown in Fig. 3.11, then the velocity - time graph is a straight line parallel to the time axis.

Fig. 3.11: Parallel to the time axis

Case 2: The graph AB represents that the body has some initial velocity AO and it is moving with uniform acceleration. The graph OC represents that the body starts from rest and it is moving with uniform acceleration as shown in Fig. 3.12.

Fig. 3.12: Uniform velocity variation

Case 3: In the Fig. 3.13, the line AB indicates that the body has initial velocity OA and it is moving with uniform retardation and finally comes to rest at point B.

Fig. 3.13: Uniform retardation velocity

Case 4: In the Fig. 3.14, it is seen that the body has initial velocity OA. The velocity decreases at uniform rate and becomes zero at point B. The body reaccelerates at uniform velocity, indicated by line segment BC.

Fig. 3.14: Uniform retardation velocity and acceleration of a body

Case 5: As seen in the Fig. 3.15, the body starts accelerating from point O and reaches to point A at uniform velocity then it decelerates (retards) at uniform velocity from point A to B and finally stops at point B.

Fig. 3.15: Uniform increase and decrease in velocity

Case 6: If the velocity-time of a graph is a curve, as seen in Fig. 3.16, then the body with motion is with variable velocity (non - uniform velocity).

Fig. 3.16: Non-uniform velocity

Case 7: In this case, the body has some initial velocity (say OA of Fig. 3.17) and is moving wit uniform velocity from A to B while B to C represents the uniform retardation. The graph CD represents motion with uniform velocity, whereas DEF represents that the body is moving with variable velocity under variable acceleration and retardation.