There are all sorts of resonances around us, in the world, in our culture, and in our technology. A tidal resonance causes the 55 foot tides in the Bay of Fundy. Mechanical and acoustical resonances and their control are at the center of practically every musical instrument that ever existed. Even our voices and speech are based on controlling the resonances in our throat and mouth. Technology is also a heavy user of resonance. All clocks, radios, televisions, and gps navigating systems use electronic resonators at their very core. Doctors use magnetic resonance imaging or MRI to sense the resonances in atomic nuclei to map the insides of their patients. In spite of the great diversity of resonators, they all share many common properties. In this blog, we will delve into their various aspects. It is hoped that this will serve both the students and professionals who would like to understand more about resonators. I hope all will enjoy the animations.

For a list of all topics discussed, scroll down to the very bottom of the blog, or click here.

Origins of Newton's laws of motion

Non-mathematical introduction to relativity

Three types of waves: traveling waves, standing waves and rotating waves new

History of mechanical clocks with animations
Understanding a mechanical clock with animations
includes pendulum, balance wheel, and quartz clocks

Water waves, Fourier analysis

Tuesday, July 12, 2011

Relativistic kinetic energy of a particle

all topics by author introduction to relativity contents of mathematics of relativity previous: relativistic mass next: transforming electromagnetic fields - contents

Einstein's equating of mass and energy through his famous equation  E = mc2  was later verified by the invention of the atomic bomb. The atomic bomb converts a small amount of mass into heat and various forms of electromagnetic radiation. It is also thought that the sun derives its energy from the continual conversion of large amounts of mass into heat and light. Many other nuclear reactions similarly convert mass into energy.

Most physicists today consider mass and energy to be attributes of the same property. That is to say, that mass and energy are always proportional to each other as given by the above equation. Thus a hot object will be slightly more massive than when it is cold, although the difference in mass is so slight as to be very hard to detect.

If the mass increases with velocity as discussed in the previous chapter, what is the relationship between kinetic energy and velocity?

Classical kinetic energy

In classical (low speed) physics we are taught that the kinetic energy of a particle is given by:

     KE = ½mv2    .    (11.1)

We can derive this as

classical derivation of kinetic energy

classical derivation of kinetic energy    .    (11.2)

where all the vectors are in the forward direction of motion (in the x direction) so we just show these as scalars. Finishing the derivation, we have:

classical derivation of kinetic energy    .

Derivation of relativistic kinetic energy

In the relativistic case, we repeat the same steps as above except that now the mass is a function of velocity as given by:

equation for relativistic mass    .    (11.4)

We now repeat the steps in (11.2) and (11.3) with this new non-constant mass that varies with velocity v. This calculation will be more laborious because of this new addition and will take a few lines of equations:

first equation in derivation of relativistic kinetic energy    ,    (11.5)

second equation in derivation of relativistic kinetic energy    ,    (11.6)

where we have used integration by parts for the second step. Also, please excuse the monster integral signs. This is one of the few flaws in the equation editor I use, MathCast, which is otherwise extremely good, especially considering that it is free, very easy to install, and makes entering equations very fast.

third equation in derivation of relativistic kinetic energy    ,    (11.7)

where u is defined as:

definition of u used in the above calculation    .    (11.8)

Continuing with (11.7), we have:

fourth equation in derivation of relativistic kinetic energy    ,    (11.9)


fifth and final equation in derivation of relativistic kinetic energy    .    (11.10)

Interpretation of relativitic energy

It is standard to call the left-most term of the final expression in (11.10) the "total relativistic energy" or Etotal and the right-most term the "relativistic rest mass energy" or E0:

total relativistic energy    ,    (11.11)

relativistic rest mass energy    .              (11.12)

Making this substitution, we write (11.10) as:

relativistic total energy equation    .    (11.13)

Interpreted this way, the total energy of a particle (or object or mass) has two components, the kinetic energy and the rest mass energy. Identifying E0 as an energy was quite a leap of faith for Einstein, because at the time there was no conceivable way to extract this energy for use or even for experimental verification that this term indeed was an "energy" i.e. an ability to do work. He made the leap primarily because it made a nice equation for the energy as shown in (11.11) above. The ensuing years revealed the annihilation of matter and antimatter, nuclear fission, and nuclear fusion, all of which converted this rest mass energy to the more normal forms of energy (light and heat) verifying his bold assumption.

Incorporating the rest mass into the energy also allowed the energy and the three momentum components (px, py, and pz) to make up a "four vector", a concept we shall briefly explain in Chapter 14.

Momentum/energy transform

We can use our new mass/energy relationship (11.11) to rewrite the momentum/mass transformation, equation (10.25), at the end of the previous chapter as:

Momentum/energy transform    ,    (11.14)

where in this equation we represent the total energy of the object (rest energy plus kinetic energy) as  E . We have also multiplied the bottom items in (10.25) on either side of the equal sign by  c2 .

all topics by author introduction to relativity contents of mathematics of relativity previous: relativistic mass next: transforming electromagnetic fields - contents