# Kinetic energy

The kinetic energy of an object is the energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work is done by the body in decelerating from its current speed to a state of rest.

The speed, and thus the kinetic energy of a single object is frame-dependent (relative): it can take any non-negative value, by choosing a suitable inertial frame of reference. For example, a bullet passing an observer has kinetic energy in the reference frame of this observer. The same bullet is stationary from the point of view of an observer moving with the same velocity as the bullet, and so has zero kinetic energy. By contrast, the total kinetic energy of a system of objects cannot be reduced to zero by a suitable choice of the inertial reference frame, unless all the objects have the same velocity. In any other case the total kinetic energy has a non-zero minimum, as no inertial reference frame can be chosen in which all the objects are stationary. This minimum kinetic energy contributes to the system's invariant mass, which is independent of the reference frame.

In classical mechanics, the kinetic energy of a non-rotating object of mass m traveling at a speed v is ½ mv². Inrelativistic mechanics, this is only a good approximation when v is much less than the speed of light

## Newtonian kinetic energy

### Kinetic energy of rigid bodies

In classical mechanics, the kinetic energy of a point object (an object so small that its mass can be assumed to exist at one point), or a non-rotating rigid body, is given by the equation

$E_k = frac{1}{2} mv^2$

where $m$ is the mass and $v$ is the speed (or the velocity) of the body. In SI units (used for most modern scientific work), mass is measured in kilograms, speed in metres per second, and the resulting kinetic energy is in joules.

For example, one would calculate the kinetic energy of an 80 kg mass (about 180 lbs) traveling at 18 metres per second (about 40 mph, or 65 km/h) as

Ek = (1/2) · 80 · 182 J = 12.96 kJ

Since the kinetic energy increases with the square of the speed, an object doubling its speed has four times as much kinetic energy. For example, a car traveling twice as fast as another requires four times as much distance to stop, assuming a constant braking force.

The kinetic energy of an object is related to its momentum by the equation:

$E_k = frac{p^2}{2m}$

where:

$p;$ is momentum
$m;$ is mass of the body

For the translational kinetic energy, that is the kinetic energy associated with rectilinear motion, of a rigid body with constant mass $m;$, whose center of mass is moving in a straight line with speed $v;$, as seen above is equal to

$E_t = frac{1}{2} mv^2$

where:

$m;$ is the mass of the body
$v;$ is the speed of the center of mass of the body.

The kinetic energy of any entity depends on the reference frame in which it is measured. However the total energy of an isolated system, i.e. one which energy can neither enter nor leave, does not change in whatever reference frame it is measured. Thus, the chemical energy converted to kinetic energy by a rocket engine is divided differently between the rocket ship and its exhaust stream depending upon the chosen reference frame. This is called the Oberth effect. But the total energy of the system, including kinetic energy, fuel chemical energy, heat, etc., is conserved over time, regardless of the choice of reference frame. Different observers moving with different reference frames disagree on the value of this conserved energy.

The kinetic energy of such systems depends on the choice of reference frame: the reference frame that gives the minimum value of that energy is the center of momentum frame, i.e. the reference frame in which the total momentum of the system is zero. This minimum kinetic energy contributes to the invariant mass of the system as a whole.

#### Derivation

The work done accelerating a particle during the infinitesimal time interval dt is given by the dot product of force and displacement:

$mathbf{F} cdot d mathbf{x} = mathbf{F} cdot mathbf{v} d t = frac{d mathbf{p}}{d t} cdot mathbf{v} d t = mathbf{v} cdot d mathbf{p} = mathbf{v} cdot d (m mathbf{v}),,$

where we have assumed the relationship p = m v. (However, also see the special relativistic derivation below.)

Applying the product rule we see that:

$d(mathbf{v} cdot mathbf{v}) = (d mathbf{v}) cdot mathbf{v} + mathbf{v} cdot (d mathbf{v}) = 2(mathbf{v} cdot dmathbf{v}).$

Therefore (assuming constant mass), the following can be seen:

$mathbf{v} cdot d (m mathbf{v}) = frac{m}{2} d (mathbf{v} cdot mathbf{v}) = frac{m}{2} d v^2 = d left(frac{m v^2}{2} ight).$

Since this is a total differential (that is, it only depends on the final state, not how the particle got there), we can integrate it and call the result kinetic energy:

$E_k = int mathbf{F} cdot d mathbf{x} = int mathbf{v} cdot d (m mathbf{v}) = int d left(frac{m v^2}{2} ight) = frac{m v^2}{2}.$

This equation states that the kinetic energy (Ek) is equal to the integral of the dot product of the velocity (v) of a body and the infinitesimal change of the body'smomentum (p). It is assumed that the body starts with no kinetic energy when it is at rest (motionless).

### Rotating bodies

If a rigid body is rotating about any line through the center of mass then it has rotational kinetic energy ($E_r,$) which is simply the sum of the kinetic energies of its moving parts, and is thus given by:

$E_r = int frac{v^2 dm}{2} = int frac{(r omega)^2 dm}{2} = frac{omega^2}{2} int{r^2}dm = frac{omega^2}{2} I = egin{matrix} frac{1}{2} end{matrix} I omega^2$

where:

• ω is the body's angular velocity
• r is the distance of any mass dm from that line
• $I,$ is the body's moment of inertia, equal to $int{r^2}dm$.

(In this equation the moment of inertia must be taken about an axis through the center of mass and the rotation measured by ω must be around that axis; more general equations exist for systems where the object is subject to wobble due to its eccentric shape).

### Office hours

Sunday:     12-1

Monday:     12-1

Tuesday:     10-12

Wednesday: 10-11

You are welcome to contact me by at any time

You  may also contact me via WhatsApp group, ..

Ext. 2524

### Civil Eng. Students

DON'T miss!

As we approaching the second midterm exams..  Get ready and remember our offices are open to answer you . do not hesitate to .contact me if you have difficulties

For CE 370 students : I do apologize for did not attend the extra lab session that proposed to be on Wednesday at 6 am.  Sorry

Good Luck

### Announcements

My Dear Students

.contact me  to sort it out

There will be an orientation session in the midell of this term to help you select your track in Civil and Environmental Engineering

### Event

#### Special Issue

Published Online on January 2016

### Mining Engineering

Mining engineering is an engineering discipline that involves practice, theory, science, technology, and the application of extracting and processing minerals from a naturally occurring environment. Mining engineering also includes processing minerals for additional value.

### Environmental Engineering

Environmental engineers are the technical professionals who identify and design solutions for    environmental problems. Environmental engineers provide safe drinking water, treat and properly dispose of wastes, maintain air quality, control water pollution, and remediate sites contaminated due to spills or improper disposal of hazardous substances. They monitor the quality of the air, water, and land. And, they develop new and improved means to protect the environment.

### Proverb

Actions speak louder than words

### Univeristies

Universities I've worked in

Assuit University (Home University), Egypt

Imperial College, London, UK

King Saud University, KSA

,Majmaah University

### Also visit

https://www.researchgate.net/profile/Sameh_Ahemd

and give me

### Think

How many red balls we need to make balance

### Sports

Egypt vs Tunisia (Handball final)  30/1/2016

### Course 2016/17-1

1. Computer Applications in Surveying  CE 473
2. Surveying 1 CE 370
3. Photogrammetry CE 474
4. Surveying II  CE 371
5. Design I  (round 4) CE 498

Rule #7

### Latest

Member of the Editorial Board: " Journal of Water Resources and Ocean Sciences"  2013

Participating in The Third International Conference on Water, Energy and Environment,(ICWEE) 2015 - American University of Sharjah, UAE 24-26 March 2015 with a Paper and Poster

New article

Recovery of Titania from Waste-Sludge of Majmaah Water Treatment Plant

"Production of Titania Nano-particles from Wast-

Sludge

### Coming soon

Results of M2 CE311

### Student Conference

Participation  in the 6Th Student Conference

With a paper and oral presentation

From the Senior Design Project CE499 -35

See inside, the paper, and presentation

## Surveying  I -  CE 370 - 2016-2017-1

 Power point Sheets Lecture notes Second Midterm Exam Results Model Answer

Student Performance Records

## Surveying II - CE 371 - 2016-2017-1

 Lectures First midterm exam Lab Results

See Inside

Student Performance Records

## CE 473

 Lecturers Power point Sheets Exams + Quizzes Results

### 2015-2016-2 - Photogrammetry  CE474

 Available 0-1-2-3-4 Power point Available Quizzes 2 and 3 with model Answer Quizzes Available Chapter 1,2,3,4,5 Lecture notes Report 1  Cameras Report Y 60 marks Exams

See Inside

Student Performance Records

## Environmental Engineering 1

CE 360: Environmental Engineering 1

37-2

 Y PP0-1-2-3,4 Power point Y 1,2,3,4,5 Chapters Water quality Poster + Climate Change Reports will be announced (Quiz #2 will be using D2L- Online Quizzes Y Quizzez 2,3 and 4 with model answer and results Exams and results

### Engineering Report Writing

GE 306: Engineering Report Writing

### Senior Design 2 - CE 499

Meeting on 14-4-2015

Second Best paper from Senior Design Projects in 2015

Paper title:

### Evaluation of Groundwater Quality Parameters using Multivariate Statistics- a case Study of Majmaah, KSA

Students:

Abdullah A. Alzeer

Husam K. Almubark

Maijd M. Almotairi

### CE 360-Summer Course

Environmental Engineering I

## Welcome to CE360 second Term 2015-2016

### Engineering Practice

Engineering Practice GE 307

### الهيئة الوطنية للاعتماد والتقويم الاكاديمي

See what can you get from Google!! more than translation and locations

### Contact me

Mobile: 00966598311652

### Thank you

Your frequent visit to my website has helped  a lot to get the 2nd place in the university competition in the  year 1434-2013... Thanks very mush and please keep following my work

زيارتكم المتكررة لموقعي ساهمت في حصولي على المركز الثاني لاحسن المواقع الشخصية بجامعة المجمعة ...لكم جميعا خالص شكري وتقديري

## CE 212

 Y from PP0 to PP 6 + PDF Power points Y Midterm #2 + Model Answer + results Exams Y lab 3 Labs Solution of Quiz 4 + Results HW

## CE 311

M2+ model Answer + PP 1-9 = PDF 1-9 all are available

Sheets 2 + 4 with model Answers

Results of M2

### إحصائية الموقع

عدد الصفحات: 1412

البحوث والمحاضرات: 1544

الزيارات: 66380