The time rate of change of velocity of an object is called acceleration of the object.
(1) It is a vector quantity. It?s direction is same as that of change in velocity (Not of the velocity)
Possible ways of velocity change
When only direction of velocity changes
When only magnitude of velocity changes
When both magnitude and direction of velocity changes
Acceleration perpendicular to velocity
Acceleration parallel or anti-parallel to velocity
Acceleration has two components one is perpendicular to velocity and another parallel or anti-parallel to velocity
Ex.. Uniform circular motion
Ex.. Motion under gravity
Ex.. Projectile motion
(2) Dimension : \[[{{M}^{0}}{{L}^{1}}{{T}^{-2}}]\]
(3) Unit : metre/second2 (S.I.); cm/second2 (C.G.S.)
(4) Types of acceleration :
(i) Uniform acceleration : A body is said to have uniform acceleration if magnitude and direction of the acceleration remains constant during particle motion.
(ii) Non-uniform acceleration : A body is said to have non-uniform acceleration, if either magnitude or direction or both of them change during motion.
(iii) Average acceleration : \[{{\vec{a}}_{a\upsilon }}=\frac{\Delta \vec{v}}{\Delta t}=\frac{{{{\vec{v}}}_{2}}-{{{\vec{v}}}_{1}}}{\Delta t}\]
The direction of average acceleration vector is the direction of the change in velocity vector as \[\vec{a}=\frac{\Delta \vec{v}}{\Delta t}\]
(iv) Instantaneous acceleration = \[\vec{a}=\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{\Delta \vec{v}}{\Delta t}=\frac{d\vec{v}}{dt}\]
(v) For a moving body there is no relation between the direction of instantaneous velocity and direction of acceleration.
Ex.. (a) In uniform circular motion \[\theta ={{90}^{o}}\] always
(b) In a projectile motion \[\theta \] is variable for every point of trajectory.
(vi) If a force \[\overrightarrow{F}\] acts on a particle of mass m, by Newton's 2nd law, acceleration \[\vec{a}=\frac{{\vec{F}}}{m}\]
(vii) By definition \[\vec{a}=\frac{d\vec{v}}{dt}=\frac{{{d}^{2}}\vec{x}}{d{{t}^{2}}}\]\[\left[ \text{As}\,\,\vec{v}=\frac{d\vec{x}}{dt} \right]\]
i.e., if x is given as a function of time, second time derivative of displacement gives acceleration
(viii) If velocity is given as a function more...
During motion of the particle its parameters of kinematical analysis (v, a, s) changes with time. This can be represented on the graph.
Position time graph is plotted by taking time t along x-axis and position of the particle on y-axis.
Let AB is a position-time graph for any moving particle
As Velocity = \[\frac{\text{Change in position}}{\text{Time taken}}=\frac{{{y}_{2}}-{{y}_{1}}}{{{t}_{2}}-{{t}_{1}}}\] ...(i)
From triangle ABC, \[\tan \theta =\frac{BC}{AC}=\frac{AD}{AC}=\frac{{{y}_{2}}-{{y}_{1}}}{{{t}_{2}}-{{t}_{1}}}\] ...(ii)
By comparing (i) and (ii) Velocity \[=\tan \theta \]
\[u=\tan \theta \]
It is clear that slope of tangent on position-time graph represents the velocity of the particle.
Various position -time graphs and their interpretation
\[\theta ={{0}^{o}}\] so \[\upsilon =0\]
i.e., line parallel to time axis represents that the particle is at rest.
\[\theta ={{90}^{o}}\] so \[\upsilon =\infty \]
i.e., line perpendicular to time axis represents that particle is changing its position but time does not changes it means the particle possesses infinite velocity. Practically this is not possible.
\[\theta =\] constant so \[\upsilon =\] constant, a = 0
i.e., line with constant slope represents uniform velocity of the particle.
\[\theta \] is increasing so v is increasing, a is positive.
i.e., line bending towards position axis represents increasing velocity of particle. It means the particle possesses acceleration.
\[\theta \] is decreasing so \[\upsilon \] is decreasing, a is negative
i.e., line bending towards more...
The graph is plotted by taking time t along x-axis and velocity of the particle on y-axis.
Calculation of Distance and displacement : The area covered between the velocity time graph and time axis gives the displacement and distance travelled by the body for a given time interval.
Total distance \[=\,|{{A}_{1}}|\,+\,|{{A}_{2}}|\,+\,|{{A}_{3}}|\,\]
= Addition of modulus of different area. i.e. \[s\,=\,\int{|\upsilon |\,dt}\]
Total displacement \[={{A}_{1}}+{{A}_{2}}+{{A}_{3}}\]
= Addition of different area considering their sign.
i.e. \[r\,=\,\int{\upsilon \,dt}\]
Area above time axis is taken as positive, while area below time axis is taken as negative
here \[{{A}_{1}}\] and \[{{A}_{2}}\] are area of triangle 1 and 2 respectively and \[{{A}_{3}}\] is the area of trapezium
Calculation of Acceleration : Let AB is a velocity-time graph for any moving particle
As Acceleration = \[\frac{\text{Change in velocity}}{\text{Time taken}}\]
\[=\frac{{{v}_{2}}-{{v}_{1}}}{{{t}_{2}}-{{t}_{1}}}\] ...(i)
From triangle ABC, \[\tan \theta =\frac{BC}{AC}=\frac{AD}{AC}\]
\[=\frac{{{v}_{2}}-{{v}_{1}}}{{{t}_{2}}-{{t}_{1}}}\] ...(ii)
By comparing (i) and (ii)
Acceleration (a) = \[\tan \theta \]
It is clear that slope of tangent on velocity-time graph represents the acceleration of the particle.
Various velocity -time graphs and their interpretation
\[\theta ={{0}^{o}},\,\,a=0,\,\,\upsilon =\] constant
i.e., line parallel to time axis represents that the particle is moving with constant velocity.
\[\theta ={{0}^{o}},\,\,a=\infty ,\,\,\upsilon =\] increasing
i.e., line perpendicular to time axis represents that the particle is increasing its velocity, but time does not change. It means the particle possesses infinite acceleration. Practically it is not possible.
\[\theta =\] constant, so a = constant and \[\upsilon \] is increasing uniformly with time
i.e., line with constant slope represents uniform acceleration of the particle.
These are the various relations between \[u,\,\,\upsilon ,\,\,a,\,\,t\] and s for the particle moving with uniform acceleration where the notations are used as :
u = Initial velocity of the particle at time t = 0 sec
\[\upsilon =\] Final velocity at time t sec
a = Acceleration of the particle
s = Distance travelled in time t sec
\[{{s}_{n}}=\] Distance travelled by the body in nth sec
(1) When particle moves with zero acceleration
(i) It is a unidirectional motion with constant speed.
(ii) Magnitude of displacement is always equal to the distance travelled.
(iii) \[\upsilon =u,\], s = u t [As a = 0]
(2) When particle moves with constant acceleration
(i) Acceleration is said to be constant when both the magnitude and direction of acceleration remain constant.
(ii) There will be one dimensional motion if initial velocity and acceleration are parallel or anti-parallel to each other.
(iii) Equations of motion Equation of motion
(in scalar from) (in vector from)
\[\upsilon =u+at\] \[\vec{v}=\vec{u}+\vec{a}t\]
\[s=ut+\frac{1}{2}a{{t}^{2}}\] \[\vec{s}=\vec{u}t+\frac{1}{2}\vec{a}{{t}^{2}}\]
\[{{\upsilon }^{2}}={{u}^{2}}+2as\] \[\vec{v}.\vec{v}-\vec{u}.\vec{u}=2\vec{a}.\vec{s}\]
\[s=\left( \frac{u+v}{2} \right)\,t\] \[\vec{s}=\frac{1}{2}(\vec{u}+\vec{v})\,t\]
\[{{s}_{n}}=u+\frac{a}{2}\,(2n-1)\] \[{{\vec{s}}_{n}}=\vec{u}+\frac{{\vec{a}}}{2}\,(2n-1)\]
The force of attraction of earth on bodies, is called force of gravity. Acceleration produced in the body by the force of gravity, is called acceleration due to gravity. It is represented by the symbol g.
In the absence of air resistance, it is found that all bodies (irrespective of the size, weight or composition) fall with the same acceleration near the surface of the earth. This motion of a body falling towards the earth from a small altitude (h << R) is called free fall.
An ideal example of one-dimensional motion is motion under gravity in which air resistance and the small changes in acceleration with height are neglected.
(1) If a body is dropped from some height (initial velocity zero)
(i) Equations of motion : Taking initial position as origin and direction of motion (i.e., downward direction) as a positive, here we have
u = 0 [As body starts from rest]
a = +g [As acceleration is in the direction of motion]
v = g t ...(i)
\[h=\frac{1}{2}g{{t}^{2}}\] ...(ii)
\[{{\upsilon }^{2}}=2gh\] ...(iii)
\[{{h}_{n}}=\frac{g}{2}(2n-1)\] ...(iv)
(ii) Graph of distance, velocity and acceleration with respect to time :
(iii) As \[h=(1/2)g{{t}^{2}},\] i.e., \[h\propto {{t}^{2}},\] distance covered in time t, 2t, 3t, etc., will be in the ratio of \[{{1}^{2}}:{{2}^{2}}:{{3}^{2}}\], i.e., square of integers.
(iv) The distance covered in the nth sec, \[{{h}_{n}}=\frac{1}{2}g\,(2n-1)\]
So distance covered in 1st, 2nd, 3rd sec, etc., will be in the ratio of 1 : 3 : 5, i.e., odd integers only.
(2) If a body is projected vertically downward with some initial velocity
Equation of motion : \[\upsilon =u+g\,t\]
\[h=ut+\frac{1}{2}g\,{{t}^{2}}\]
\[{{\upsilon }^{2}}={{u}^{2}}+2gh\] \[{{h}_{n}}=u+\frac{g}{2}\,(2n-1)\]
(3) If a body is projected vertically upward (i) Equation of motion :
Taking initial position as origin and direction of motion (i.e., vertically up) as positive
a = - g [As acceleration is downwards while motion upwards]
So, if the body is projected with velocity u and after time t it reaches up to height h then
\[\upsilon =u-g\,t\];\[h=ut-\frac{1}{2}g\,{{t}^{2}}\];\[{{\upsilon }^{2}}={{u}^{2}}-2gh\];\[{{h}_{n}}=u-\frac{g}{2}\,(2n-1)\]
(ii) For maximum height u = 0
So from above equation u = gt,
\[h=\frac{1}{2}g{{t}^{2}}\]
and \[{{u}^{2}}=2gh\]
(iii) Graph of displacement, velocity and acceleration with respect to time (for maximum height) :
It is clear that both quantities do not depend upon the mass of the body or we can say that in absence of air resistance, all bodies fall on the surface of the earth with the same rate.
(4) The motion is independent of the mass of the body, as in any equation of motion, mass is not involved. That is why a heavy and light body when released from the same height, reach the more...
(i) If acceleration is a function of time
\[a=f(t)\] then \[v=u+\int_{\,0}^{\,t}{f(t)\,dt}\]
and \[s=ut+\int_{0}^{t}{\left( \,\int{f(t)\,dt} \right)}\,dt\]
(ii) If acceleration is a function of distance
\[a=f(x)\] then \[{{v}^{2}}={{u}^{2}}+2\int_{\,{{x}_{0}}}^{\,x}{f(x)\,dx}\]
(iii) If acceleration is a function of velocity
\[a=f(\upsilon )\] then \[t=\int_{\,u}^{\,v}{\frac{dv}{f(v)}}\] and \[x={{x}_{0}}+\int_{\,u}^{\,v}{\,\frac{vdv}{f(v)}}\]
(1) Length
(i) 1 fermi = 1 fm = 10-15 m
(ii) 1 X-ray unit = 1XU = 10-13 m
(iii) 1 angstrom = 1Å = 10-10 m = 10-8 cm = 10-7 mm = 0.1 \[\mu \]mm
(iv) 1 micron = \[\mu \]m = 10-6 m
(v) 1 astronomical unit = 1 A.U. = 1. 49 x 1011 m = 1.5 x 1011 m \[\approx \] 108 km
(vi) 1 Light year = 1 ly = 9.46 x 1015 m
(vii) 1 Parsec = 1pc = 3.26 light year
(2) Mass
(i) Chandra Shekhar unit : 1 CSU = 1.4 times the mass of sun = 2.8 x 1030 kg
(ii) Metric tonne : 1 Metric tonne = 1000 kg
(iii) Quintal : 1 Quintal = 100 kg
(iv) Atomic mass unit (amu) : amu = 1.67 x 10-27 kg
Mass of proton or neutron is of the order of 1 amu
(3) Time
(i) Year : It is the time taken by the Earth to complete 1 revolution around the Sun in its orbit.
(ii) Lunar month : It is the time taken by the Moon to complete 1 revolution around the Earth in its orbit.
1 L.M. = 27.3 days
(iii) Solar day : It is the time taken by Earth to complete one rotation about its axis with respect to Sun. Since this time varies from day to day, average solar day is calculated by taking average of the duration of all the days in a year and this is called Average Solar day.
1 Solar year = 365.25 average solar day
or average solar day \[=\frac{1}{365.25}\] the part of solar year
(iv) Sedrial day : It is the time taken by earth to complete one rotation about its axis with respect to a distant star.
1 Solar year = 366.25 Sedrial day
= 365.25 average solar day
Thus 1 Sedrial day is less than 1 solar day.
(v) Shake : It is an obsolete and practical unit of time.
1 Shake = 10? 8 sec
(1) “A gel is a colloidal system in which a liquid is dispersed in a solid.”
(2) The lyophilic sols may be coagulated to give a semisolid jelly like mass, which encloses all the liquid present in the sol. The process of gel formation is called gelation and the colloidal system formed called gel.
(3) Some gels are known to liquify on shaking and reset on being allowed to stand. This reversible sol-gel transformation is called thixotropy.
(4) The common examples of gel are gum arabic, gelatin, processed cheese, silicic acid, ferric hydroxide etc.
(5) Gels may shrink by loosing some liquid help them. This is known as synereises or weeping.
(6) Gels may be classified into two types
(i) Elastic gels : These are the gels which possess the property of elasticity. They readily change their shape on applying force and return to original shape when the applied force is removed. Common examples are gelatin, agar-agar, starch etc.
(ii) Non-elastic gels : These are the gels which are rigid and do not have the property of elasticity. For example, silica gel.
“The colloidal systems in which fine droplets of one liquid are dispersed in another liquid are called emulsions the two liquids otherwise being mutually immiscible.” or
“Emulsion are the colloidal solutions in which both the dispersed phase and the dispersion medium are liquids.”
A good example of an emulsion is milk in which fat globules are dispersed in water. The size of the emulsified globules is generally of the order of \[{{10}^{-6}}\]m. Emulsion resemble lyophobic sols in some properties.
(1) Types of Emulsion : Depending upon the nature of the dispersed phase, the emulsions are classified as;
(i) Oil-in-water emulsions (O/W) : The emulsion in which oil is present as the dispersed phase and water as the dispersion medium (continuous phase) is called an oil-in-water emulsion. Milk is an example of the oil-in-water type of emulsion. In milk liquid fat globules are dispersed in water. Other examples are, vanishing cream etc.
(ii) Water-in-oil emulsion (W/O) : The emulsion in which water forms the dispersed phase, and the oil acts as the dispersion medium is called a water-in-oil emulsion. These emulsion are also termed oil emulsions. Butter and cold cream are typical examples of this types of emulsions. Other examples are cod liver oil etc.
(2) Properties of emulsion
(i) Emulsions show all the characteristic properties of colloidal solution such as Brownian movement, Tyndall effect, electrophoresis etc.
(ii) These are coagulated by the addition of electrolytes containing polyvalent metal ions indicating the negative charge on the globules.
(iii) The size of the dispersed particles in emulsions in larger than those in the sols. It ranges from 1000 Å to 10,000 Å. However, the size is smaller than the particles in suspensioins.
(iv) Emulsions can be converted into two separate liquids by heating, centrifuging, freezing etc. This process is also known as demulsification.
(3) Applications of emulsions
(i) Concentration of ores in metallurgy
(ii) In medicine (Emulsion water-in-oil type)
(iii) Cleansing action of soaps.
(iv) Milk, which is an important constituent of our diet an emulsion of fat in water.
(v) Digestion of fats in intestine is through emulsification.
Lyophilic sols are more stable than lyophobic sols.
Lyophobic sols can be easily coagulated by the addition of small quantity of an electrolyte.
When a lyophilic sol is added to any lyophobic sol, it becomes less sensitive towards electrolytes. Thus, lyophilic colloids can prevent the coagulation of any lyophobic sol.
“The phenomenon of preventing the coagulation of a lyophobic sol due to the addition of some lyophilic colloid is called sol protection or protection of colloids.”
The protecting power of different protective (lyophilic) colloids is different. The efficiency of any protective colloid is expressed in terms of gold number.
Gold number : Zsigmondy introduced a term called gold number to describe the protective power of different colloids. This is defined as, “weight of the dried protective agent in milligrams, which when added to 10 ml of a standard gold sol (0.0053 to 0.0058%) is just sufficient to prevent a colour change from red to blue on the addition of 1 ml of 10 % sodium chloride solution, is equal to the gold number of that protective colloid.”
Thus, smaller is the gold number, higher is the protective action of the protective agent.
\[\text{Protective}\,\text{power}\propto \frac{\text{1}}{\text{Gold}\,\text{number}}\]
Gold numbers of some hydrophilic substances
Hydrophilic substance
Gold number
Hydrophilic substance
Gold number
Gelatin
0.005 - 0.01
Sodium oleate
0.4 – 1.0
Sodium caseinate
0.01
Gum tragacanth
2
Hamoglobin
0.03 – 0.07
Potato starch
25
Gum arabic
0.15 – 0.25
Congo rubin number : Ostwald introduced congo rubin number to account for protective nature of colloids. It is defined as “the amount of protective colloid in milligrams which prevents colour change in 100 ml of 0.01 % congo rubin dye to which 0.16 g equivalent of KCl is added.”
Mechanism of sol protection
(i) The actual mechanism of sol protection is very complex. However it may be due to the adsorption of the protective colloid on the lyophobic sol particles, followed by its solvation. Thus it stabilises the sol via solvation effects.
(ii) Solvation effects contribute much towards the stability of lyophilic systems. For example, gelatin has a sufficiently strong affinity for water. It is only because of the solvation effects more...
Ex.. (a) In uniform circular motion \[\theta ={{90}^{o}}\] always
(b) In a projectile motion \[\theta \] is variable for every point of trajectory.
(vi) If a force \[\overrightarrow{F}\] acts on a particle of mass m, by Newton's 2nd law, acceleration \[\vec{a}=\frac{{\vec{F}}}{m}\]
(vii) By definition \[\vec{a}=\frac{d\vec{v}}{dt}=\frac{{{d}^{2}}\vec{x}}{d{{t}^{2}}}\]\[\left[ \text{As}\,\,\vec{v}=\frac{d\vec{x}}{dt} \right]\]
i.e., if x is given as a function of time, second time derivative of displacement gives acceleration
(viii) If velocity is given as a function more... 
Let AB is a position-time graph for any moving particle
As Velocity = \[\frac{\text{Change in position}}{\text{Time taken}}=\frac{{{y}_{2}}-{{y}_{1}}}{{{t}_{2}}-{{t}_{1}}}\] ...(i)
From triangle ABC, \[\tan \theta =\frac{BC}{AC}=\frac{AD}{AC}=\frac{{{y}_{2}}-{{y}_{1}}}{{{t}_{2}}-{{t}_{1}}}\] ...(ii)
By comparing (i) and (ii) Velocity \[=\tan \theta \]
\[u=\tan \theta \]
It is clear that slope of tangent on position-time graph represents the velocity of the particle.
Various position -time graphs and their interpretation
here \[{{A}_{1}}\] and \[{{A}_{2}}\] are area of triangle 1 and 2 respectively and \[{{A}_{3}}\] is the area of trapezium
Calculation of Acceleration : Let AB is a velocity-time graph for any moving particle
As Acceleration = \[\frac{\text{Change in velocity}}{\text{Time taken}}\]
\[=\frac{{{v}_{2}}-{{v}_{1}}}{{{t}_{2}}-{{t}_{1}}}\] ...(i)
From triangle ABC, \[\tan \theta =\frac{BC}{AC}=\frac{AD}{AC}\]
\[=\frac{{{v}_{2}}-{{v}_{1}}}{{{t}_{2}}-{{t}_{1}}}\] ...(ii)
By comparing (i) and (ii)
Acceleration (a) = \[\tan \theta \]
It is clear that slope of tangent on velocity-time graph represents the acceleration of the particle.
Various velocity -time graphs and their interpretation
u = 0 [As body starts from rest]
a = +g [As acceleration is in the direction of motion]
v = g t ...(i)
\[h=\frac{1}{2}g{{t}^{2}}\] ...(ii)
\[{{\upsilon }^{2}}=2gh\] ...(iii)
\[{{h}_{n}}=\frac{g}{2}(2n-1)\] ...(iv)
(ii) Graph of distance, velocity and acceleration with respect to time :
(iii) As \[h=(1/2)g{{t}^{2}},\] i.e., \[h\propto {{t}^{2}},\] distance covered in time t, 2t, 3t, etc., will be in the ratio of \[{{1}^{2}}:{{2}^{2}}:{{3}^{2}}\], i.e., square of integers.
(iv) The distance covered in the nth sec, \[{{h}_{n}}=\frac{1}{2}g\,(2n-1)\]
So distance covered in 1st, 2nd, 3rd sec, etc., will be in the ratio of 1 : 3 : 5, i.e., odd integers only.
(2) If a body is projected vertically downward with some initial velocity
Equation of motion : \[\upsilon =u+g\,t\]
\[h=ut+\frac{1}{2}g\,{{t}^{2}}\]
\[{{\upsilon }^{2}}={{u}^{2}}+2gh\] \[{{h}_{n}}=u+\frac{g}{2}\,(2n-1)\]
(3) If a body is projected vertically upward (i) Equation of motion :
Taking initial position as origin and direction of motion (i.e., vertically up) as positive
a = - g [As acceleration is downwards while motion upwards]
So, if the body is projected with velocity u and after time t it reaches up to height h then
\[\upsilon =u-g\,t\];\[h=ut-\frac{1}{2}g\,{{t}^{2}}\];\[{{\upsilon }^{2}}={{u}^{2}}-2gh\];\[{{h}_{n}}=u-\frac{g}{2}\,(2n-1)\]
(ii) For maximum height u = 0
So from above equation u = gt,
\[h=\frac{1}{2}g{{t}^{2}}\]
and \[{{u}^{2}}=2gh\]
(iii) Graph of displacement, velocity and acceleration with respect to time (for maximum height) :
It is clear that both quantities do not depend upon the mass of the body or we can say that in absence of air resistance, all bodies fall on the surface of the earth with the same rate.
(4) The motion is independent of the mass of the body, as in any equation of motion, mass is not involved. That is why a heavy and light body when released from the same height, reach the
(i) The actual mechanism of sol protection is very complex. However it may be due to the adsorption of the protective colloid on the lyophobic sol particles, followed by its solvation. Thus it stabilises the sol via solvation effects.
(ii) Solvation effects contribute much towards the stability of lyophilic systems. For example, gelatin has a sufficiently strong affinity for water. It is only because of the solvation effects
