Spirituality, Quantum Physics & Life: Kinematics

Energy is a central concept in Physics. It can be defined as the capacity for doing work. But, it is important to understand that just because energy exists, not necessarily available to do work.

Energy can not be create or destroyed, but it can change from one form to another form.

There are many types of energy such as: Mechanical energy, chemical energy, light energy, thermal energy, electrical energy etc. Mechanical energy further divided into two parts called: Kinetic Energy and Potential Energy. Here, we mainly focused on these two energy.

Kinetic Energy:

Kinetic energy is the energy of motion. It is a capacity of doing work due to its motion. There are many forms of kinetic energy - vibrational (energy due to vibrational motion), rotational (energy due to motion in circle) and translation (energy due to motion in straight line). Here, we will focus upon translation kinetic energy to keep matters simple.

In classical mechanics, the kinetic energy of an (translation) object has depends upon mass (m) and velocity (v) of an object. i.e.

$K.E. = \frac{1}{2}mv^2$

In relativistic mechanics, this is a good approximation only when $v$ is much less than the speed of light.

The amount of kinetic energy an object has depends on the mass and speed of the object.

All moving object have the kinetic energy. The faster object has more speed and has more kinetic energy.

» Calculation of Kinetic Energy:

Consider a body of mass $m$ lying on a horizontal surface. After applying a force $F$, the body travel a distance $d$ with a moving velocity $v$.

As we know from the definition of work:

W = force * distance = $F * d$ .......... (i)

From the Newton's law of motion:

$F = ma$ .......... (ii)

And from the Kinematics, we know the formula:

$v^2 = u^2 + 2ad$ .......... (iii)

If the initial velocity of a body is at rest, so the initial velocity is zero ($u = 0$), then equation (iii) becomes,

$ad = \frac{v^2}{2}$ .......... (iv)

From equation (i), (ii) and (iii), we have

$W = F * d = ma *d $

$= m * ad = m * \frac{v^2}{2}$

Thus,

∴ $W = K.E. = \frac{1}{2}mv^2$ .......... (v)

Where, m - mass and v = velocity of a body.

This equation reveals that the kinetic energy of an object is directly proportional to the square of its speed. That means, for a two fold increase in speed, the kinetic energy will increase by the factor four and so on.

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E1.4 Dynamics (Laws of Motion):

As there is no absolute frame of reference, absolute motion cannot be determined. Thus, everything in the universe can be considered to be moving (wikipedia).

In previous chapter, we have studied motion in terms of their position, velocity and acceleration. Variation of these quantities with time is called Kinematics.

In this chapter, we will discuss the cause of Motion due to the force (i.e. force acting in moving objects). This aspect of motion is called Dynamics.

In Physics, Motion is a change in position of an object over time. Motion of a body is observed by attaching a frame of reference to an observer and measuring the change in position of the body relative to that frame (wikipedia). If the position of a body is not changing with respect to a given frame of reference, the body is said to be rest (motionless).

Motion is a very common experience in our daily life. For Example: walking of people, flow of water, flying of aeroplane, running of bus etc. In physics, Motion of all large scale (i.e. Projectiles, Cells, Humans, Planets, Universe) are described by Classical Mechanics, where as motion of very small scale (i.e. atomic and subatomic objects) are described by Quantum Mechanics

Types of Motions:

Simple Harmonic Motion (Example: Pendulum), Anharmonic Motion, Periodic Motion, Circular Motion (Example: Moon and Earth), Linear Motion, Reciprocal Motion, Random Motion, Rotary Motion, Brownian Motion,Curvilinear Motion, Rotational Motion, Rolling Motion, Oscillation, Vibrational Motion, Projectile Motion, etc........

» Force and Inertia

» Newton's Laws of Motion

» Linear Momentum and Conservation laws

» Impulse of force

» Solid friction

» Application of Newton's Laws.

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E1.3 Kinematics (Projectile Motion):

In topic Kinematics, previously we almost talk about one dimensional motion. Now from this session, let's start to talk about two dimensional motion - called Projectile motion. In such type of motion, gravity is the only factor acting on our objects (neglecting the air resistance, acceleration due to gravity is downward and constant).

Any object thrown into atmosphere so that it falls under the effect of gravity alone is called Projectile motion.

Projectile motion: is a form of motion in which an object (particle) i.e. called a projectile is thrown near the Earth's surface, and it moves along a curved path under the action of gravity only. (wikipedia)

We have different types of Projectile such as:

I) Stone thrown horizontally
II) A bomb dropped from an airplane
III) Any object dropped from the windows of a Moving train.

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E1.3 Kinematics (Relative Velocity):

The velocity of one body with respect to another body is called relative velocity.

Determination the Relative Velocity as follows:

I) When two objects are moving in the same direction:
When two bodies $A$ and $B$ are moving in the same direction with velocity $V_A$ and $V_B$ respectively. Then relative velocity of $A$ with respect to $B$ is:

$V_{AB} = V_A - V_B$

Also, the relative velocity of $B$ with respect to $A$ is:

$V_{BA} = V_B - V_A$

II) When two objects are moving in the opposite direction:

When two bodies $A$ and $B$ are moving in the opposite direction with velocity $V_A$ and $V_B$ respectively. Then relative velocity of $A$ with respect to $B$ is:

$V_{AB}$ = $V_A - (-V_B)$ = $V_A + V_B$

III) When two bodies are moving inclined to each other:

When two bodies $A$ and $B$ are moving with velocities $V_A$ and $V_B$ in different directions making an angle $\theta$.

To find the velocity of B with respect to A, taking a velocity of A is reversed then taking $V_B$ and $-V_A$ as two sides of a parallelogram. The resultant gives the relative velocity as shown in figure.

(Similarly for relative velocity of A with respect to B).

Example: Rain falling vertically downward
Relative Velocity - ppt Slide

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E1.3 Kinematics (Motion of a Freely falling Body):

A freely falling object is an object that moves under the influence of gravity only.

An objects experience free fall when there is no air resistance or when air resistance is negligible.

Source

Free fall is the motion of a object where its weight is the only force acting on an object.

If neglecting air resistance (and friction), all objects in free fall in the earth's gravitational field have a constant acceleration, independent of mass, which is directed towards the earth's center (or perpendiculr to the earth's surface) and of magnitude $|\underset{a}{\rightarrow}|$ ≡ $g$ = 9.8$/s^2$.

There are two important motion characteristics that are true of free - falling objects:

Free - falling objects do not encounter air resistance (and friction).

All free - falling objects accelerate downwards at a rate of $9.8 m/s^2$.

Source

Example: If we drop a feather and a piece of rock in a tube, the rock will accelerate faster and get into the ground faster (Because, of air resistance). But if we put the feather and rock in a evacuated (removed the air) tube, both will have the same acceleration, and both get into the ground at a same time.

Equation of Motion in Straight Line

Equation of motion under gravity

For downward motion

For upward motion

$v = u + at$

$v= u +gt$

$v= u - gt$

$s = ut + \frac{1}{2} at^2$

$s = ut + \frac{1}{2} gt^2$

$s = ut - \frac{1}{2} gt^2$

$v^2 + u^2 = 2as$

$v^2 + u^2 = 2gs$

$v^2 + u^2 = - 2gs$

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E1.3 Kinematics (Equation of Motion in Straight Line):

For an object moving in a straight line, displacement, velocity, acceleration and time are taken are related by simple equation called Kinematical Equations (or equations of motion in straight line).

I. Analytical Treatment:

a) Distance covered with Uniform Velocity: $s$ = $ut$

b) Velocity of a Uniformly accelerated body after time (t): $v$ = $u$ + $at$

c) Distance covered by a uniformly accelerated body in time (t): $s$ = $ut + 1/2at^2$

d) Velocity of Uniformly accelerated body after covering a distance (S): $v^2$ - $u^2$ = $2as$

e) Distance traveled in $n^{th}$ second: $s_{n^{(th)}}$ = $u + 1/2 a(2n - 1)$

II. Graphical Treatment:

a) v = u + at

Acceleration = slope of the velocity-time graph AB

∴ a = $\frac {BM}{AM}$ = $\frac {BM}{ON}$ = $\frac {BN - MN}{ON}$ = $\frac {v - u}{t}$

or, $v - u = at$

⇒ $v = u + at$ .................... (i)

b) $s = ut + \frac{1}{2}at^2$

Acceleration = slope of the velocity-time graph AB

∴ a = $\frac{BM}{AM}$ = $\frac{BM}{t}$; ⇒ $BM$ = $at$

Now, distance traveled by object in time t,

s = area of trapezium OABN = area of rectangle OAMN + area of triangle ABM

= OA * ON + $\frac{1}{2}OB$ * AM = ut + $\frac{1}{2}at$ * t = ut + $\frac{1}{2}at^2$

Therefore, s = ut + $\frac{1}{2}at^2$ .............. (ii)

c) $v^2$ = $u^2$ + 2as

From the velocity - time graph:

s = area OABN

= $\frac{1}{2}(OA + NB).AM$

= $\frac{1}{2}(OA + NB). \frac{AM}{BM}.BM$

= $\frac{1}{2}(u + v). \frac{1}{BM/AM}.(BN - MN)$

= $\frac{1}{2}(u + v)\frac{1}{a}(v - u)$

= $\frac{v^2 - u^2}{2a}$

∴ $v^2 - u^2 = 2as$ .......................... (iii)

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E1.3 Kinematics (Acceleration):

If an object is moving with changing velocity (either increasing or decreasing), the body is said to be in accelerated motion.

Where, V = Final Velocity, U = Initial Velocity and t = Time taken

Acceleration: In Physics, the rate of change of velocity of an object with respect to time. It is a vector quantity. It's unit is $m/s^2$. To calculated net force is equal to the product of the object's mass and it's acceleration, as described by Newton's Second Law (i.e. $ F$ = $ma$, Where, F = Force, m = Mass and a = Acceleration).

For example: When a car starts from a rest (zero relative velocity) and travels in a straight line at increasing speeds. It is accelerating in the direction of travel called Positive acceleration. If the speed of the car decreases, this is an acceleration in the opposite direction from the direction of the car, it is called the negative acceleration (deceleration).

i) Average Acceleration: Ratio of total change in velocity to the total time interval. This is independent of the velocities at different points of the path but depends only on the velocities at time $t_1$ and $t_2$.

i.e. $a_{av}$ = $\frac {v_2 - v_1}{t_2 - t_1}$

ii) Instantaneous Acceleration: Acceleration is calculated for that particular point of the path of motion at the instant. This gives instantaneous acceleration. And is give by:

Instantaneous Acceleraton ($a$) = $\lim_{\Delta t \rightarrow 0} \frac{\Delta v}{\Delta t}$

This instantaneous acceleration is also called 'acceleration'. It is a vector quantity ($m/s^2$).

Types of Acceleration:

i) Positive Acceleration: Velocity goes on increasing, acceleration is Positive. In +ve acceleration direction of acceleration is in the direction of velocity.

ii) Negative Acceleration: When velocity goes on decreasing, acceleration is negative. It is also called deceleration (retardation). Direction of velocity is opposite to the acceleration direction.

iii) Zero Acceleration: It is also called Uniform velocity. When velocity does not change and remain uniform thus there is no acceleration.

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E1.3 Kinematics (Speed and Velocity):

1. Speed: Speed is a scalar quantity. This is a distance traveled by an object per unit time. Speed does not depend on the direction but on the total length in any direction. The unit of speed is $m/s$.

If $d$ is the distance traveled in time $t$, then speed $(S)$ = $\frac{distance\; (d)}{time \;(t)}$.

Picture Source

(i) Average Speed: total distance traveled by the object in any direction in one second. It is defined as the rate of change of distance of the object from the fixed point. It's unit is $m/s$.

Let $S$ be the total distance traveled by the object in time interval $t_1$ to $t_2$.

i.e. $S_{av}$ = $\frac{S}{t_2 - t_1}$ (∵ ${t_2 - t_1}$ = $ \Delta t$).

(ii) Instantaneous Speed: The speed of the object at a particular moment.It's unit is $m/s$. Let $\Delta S$ be the very small distance travelled by an object in a very short time interval $\Delta t$, approaches to zero.

i.e. Instantaneous Speed ($S$) = $\lim_{\Delta t\rightarrow 0}$ = $\frac{\Delta S}{\Delta t}$

2. Velocity: Velocity is a Vector quantity. This is a Displacement of the body per unit time. Velocity depends on the direction of motion. The unit of velocity is $m/s$.

Let $D$ be the displacement traveled in time $t$, then velocity $(V)$ = $\frac{D}{t}$. [i.e. $\frac{Distance\; + \;Direction}{time}$].

Picture Source

(i) Average Velocity: It is defined as the ratio of its total displacement to the total time interval in which the displacement occurs.

$S_{av}$ = $\frac{S_2 - S_1}{t_2 - t_1}$ = $\frac{\Delta S}{\Delta t}$.

(∵ $ \Delta S$ = total displacement and $ \Delta t$ = total time)

(ii) Instantaneous Velocity: The velocity of an object at any given instant of time at particular point of its path is called instantaneous velocity.

Instantaneous Velocity($S$) = $\lim_{\Delta t\rightarrow 0}$ = $\frac{\Delta S}{\Delta t}$

Uniform and Non-Uniform Velocity:

Uniform Velocity: If a body travels equal distances in equal intervals of time.

Non-Uniform Velocity: If an object does not travel equal distances in equal intervals of time or the direction of motion changes, the body is said to be in non-uniform (Variable) velocity.

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E1.3 Kinematics (Distance and Displacement):

Motion of an object is defined as the change in position of the object with respect to its surroundings. If the object does not change their position with respect to their surroundings, which is said to be at rest.

Distance: Distance is a scalar quantity. The length of actual path followed by an object between its initial and final position is called distance. There is no directional component to a distance measurement.

Displacement: Displacement is a vector quantity. It is independent of actual path followed by an object but depends only on the distance between initial and final position.

This is a best example to visualize what's the difference between Distance and Dispalcement.

Calculating Distance and Displacement:

Here I want to illustrate the concept of distance and displacement.

From the Figure, If I walk $3m$ west and turn to $4m$ north. Then easily to calculate the distance what I traveled.

i.e. $3m$ + $4m$ = $7m$.

This $7m$ is just only a distance what I traveled and no need to explain the direction of travel.

But, to determine Displacement; first of all I can draw an arrow where I started walking and extends to the point where I stopped walking.
From the Pythagorean Theorem: $a{^2}$ + $b{^2}$ = $c{^2}$. Since $c^2$ is our displacement squared; it should be written as: $(\Delta S){^2}$.

⟹ $(\Delta S)$ = $\sqrt{3^2 + 4^2}$ = $\sqrt{9 + 16}$ = $\sqrt{25}$ = $5$

So, the displacement is $5m$ north west from the initial position. Where need to specify the direction to explain the displacement.

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Equation of Motion in Straight Line	Equation of motion under gravity
Equation of Motion in Straight Line	For downward motion	For upward motion
$v = u + at$	$v= u +gt$	$v= u - gt$
$s = ut + \frac{1}{2} at^2$	$s = ut + \frac{1}{2} gt^2$	$s = ut - \frac{1}{2} gt^2$
$v^2 + u^2 = 2as$	$v^2 + u^2 = 2gs$	$v^2 + u^2 = - 2gs$