Today I thought I would blog about another of my favourite Meic Stevens songs, “Dic Penderyn”. This song appears on his 1972 album Gwymon, and also on his compilation album Disgwyl Rhywbeth Gwell i Ddod.

The subject matter of this song is the Welsh labourer and coal miner Richard Lewis (1807-1831). He has become better known as Dic Penderyn, and he has gone down in Welsh history as being one of the heroes of the Merthyr Rising on 3 June 1831. This was a day of violence which capped many years of unrest amongst the working class people of Merthyr Tydfil in South Wales, after their wages had been cut and the unemployment rate had risen.

Dic Penderyn was charged with stabbing a soldier with a bayonet during the riot. Despite the people of Merthyr Tydfil doubting his guilt (they even signed a petition for his release), he was found guilty and hanged on 13 August 1831 outside Cardiff Castle.

Here is the YouTube video of Meic Stevens’ song about Dic Penderyn.

Here are the lyrics to this song.

Dic Penderyn, wyt ti’n foi

Ble est ti’n yfed was?

Lawr yn Merthyr oeddwn ddoe

O saith tan hanner nos.

Pwy sy’n gweithio yn y pwll?

Pwy sy’n yfed medd?

Mae’r Cymry’n bwyta bara sych,

Mae’n dywyll fel y bedd.

Dic Penderyn, redwch nawr

Mae’r milwyr ar dy ôl.

Cotiau coch a’u gynniau gwyllt

Saethu bobol ffôl.

Saethu bobol ffôl.

Pwy sy’n gweithio yn y pwll?

A pwy sy’n yfed medd?

Mae’r Cymry’n bwyta bara sych,

Mae’n dywyll fel y bedd.

Dic Penderyn, cuddiwch nawr

Mae plismyn yn ein stryd.

Dewch mâs yn gloi trwy drws y cefn

Rhowch ‘ goesau di yn rhydd

Ie, tra bo’ch goesau di yn rhydd.

Pwy sy’n gweithio yn y pwll?

Pwy sy’n yfed medd?

Mae’r Cymry’n bwyta bara sych,

Mae’n dywyll fel y bedd.

O, mae’n dywyll fel y bedd.

Dic Penderyn, o rhy hwyr

Mae e’n hedfan dros y bryn.

Ar llawr y carchar mae o nawr

A’r ddial rownd pob glun.

O, a’r haearn rownd pob glun.

Pwy sy’n gweithio yn y pwll?

Pwy sy’n yfed medd?

Mae’r Cymry’n bwyta bara sych,

Mae’n dywyll fel y bedd.

O, mae’n dywyll fel y bedd.

Dic Penderyn, Cymro glân

Llofruddwr nawr wyt ti.

Wrth borth y castell yng Nghaerdydd

Rhaff y Sais gei di.

Ie, rhaff y Sais gei di.

Pwy sy’n gweithio yn y pwll?

Pwy sy’n yfed medd?

Mae’r Cymry’n bwyta bara sych,

Mae’n dywyll fel y bedd.

O, mae’n dywyll fel y bedd.

Ie, mae’n dywyll fel y bedd.

As usual, my translation is as accurate to the Welsh lyrics as I can make it, without any attempt to retain a rhythm or rhyme in the English words.

Dic Penderyn, you are a boy

Where did you go drinking son?

I was down in Merthyr yesterday

From seven until midnight.

Who works in the mine?

Who drinks mead?

The Welsh people eat dry bread,

It’s as dark as a grave.

Dic Penderyn, run now

The soldiers are after you.

Red coats and wild rifles

Shooting crazy people.

Shooting crazy people.

Who works in the mine?

Who drinks mead?

The Welsh people eat dry bread,

It’s as dark as a grave.

Dic Penderyn, hide now

The policemen are in our street.

Come out quickly through the back door

Let your legs be free

Yes, whilst your legs are free.

Who works in the mine?

Who drinks mead?

The Welsh people eat dry bread,

It’s as dark as a grave.

Oh, it’s as dark as a grave.

Dic Penderyn, oh it’s too late

He is flying over the hill.

On the floor of the jail he is now

With revenge around each hip.

Oh, with the iron around each hip.

Who works in the mine?

Who drinks mead?

The Welsh people eat dry bread,

It’s as dark as a grave.

Oh, it’s as dark as a grave.

Dic Penderyn, pure Welshman

You are now a murderer.

At the gates of Cardiff Castle

You’ll get the Englishman’s rope.

Yes, you’ll get the Englishman’s rope.

Who works in the mine?

Who drinks mead?

The Welsh people eat dry bread,

It’s as dark as a grave.

Oh, it’s as dark as a grave.

Yes, it’s as dark as a grave.

5. Simple Twist Of Fate (1975)

Dylan’s saddest song. As he sings of the last night of a couple together with weary despair, Dylan’s narrative unfolds in the third person, except for one devastating giveaway “I remember well”. Even the wheezy harmonica solo sounds emotionally shattered.

4. Jokerman (1983)

Many of Dylan’s most devoted fans were alienated by the preachiness of Dylan’s born again Christian phase. On Jokerman he released himself back into a beautiful ambiguity that more perfectly distils the mysteries of faith. The Jokerman is Jesus, “born with a snake in both of your fists while a hurricane was blowing.” With legendary Jamaican rhythm section Sly Dunbar and Robbie Shakespeare pulsing beneath Mark Knopfler’s silvery guitar, the track has a slipperiness that mirrors its audacious lyrical twists and turns.

3. A Hard Rain’s A Gonna Fall (1963)

Adapting the melody and refrain of traditional English folk song Lord Randall, Dylan lets loose the full force of his poetic imagination like an apocalyptic storm. This first recording is sparse, just strummed acoustic guitar and that barbed wire voice, but the cascade of imagery holds listeners in hypnotic thrall.

2. Blowin’ In the Wind (1963)

This song was released in 1962 by Dylan on his Freewheelin’ Bob Dylan album.

The song that really announced Dylan, a troubadour for a generation, old beyond his years. He was just 21 when he wrote it but cautiously held it back from his debut album of folk covers. It sounds like a song that has been blowing around for 1000 years and has a quality of simple, universal, mysterious truth that will keep it relevant for a thousand more.

1. Tangled Up In Blue (1975)

The most dazzling lyric ever written, an abstract narrative of relationships told in an amorphous blend of first and third person, rolling past, present and future together, spilling out in tripping cadences and audacious internal rhymes, ripe with sharply turned images and observations and filled with a painfully desperate longing. “I wanted to defy time” according to Dylan. “When you look at a painting, you can see any part of it altogether. I wanted that song to be like a painting.”

In this blogpost here, I derived the Lorentz transformations from first principles. The derivation uses a simple thought experiment; two reference frames S \text{ and } S^{\prime} are moving relative to each other with a speed v. The origins of S \text{ and } S^{\prime} coincide spatially at time t=t^{\prime}=0. At this moment, a flash of light is created at their origins, and expands as a sphere.

The radius of this sphere of light will be r \text{ and } r^{\prime} in S \text{ and } S^{\prime} respectively. So, we can write

x^{2} + y^{2} + z^{2} = r^{2} \text{ in } S \text{ (1) }
x^{\prime}{^2} + y^{\prime}{^2} +z^{\prime}{^2} = r^{\prime}{^2} \text{ in } S^{\prime} \text{ (2) }

But, in reference frame S, the distance r that light travels in time t is related to the speed of light c, as c = r/t, so for S we can write that

r = ct

We can do the same thing for reference frame S^{\prime}. The distance r^{\prime} that light travels in time t^{\prime} in S^{\prime} is related to the speed of light c^{\prime} in S^{\prime}, c^{\prime} = r^{\prime} / t^{\prime}, so for S^{\prime} we can write

r^{\prime} = c^{\prime} t^{\prime}.

However, crucially, Einstein said that the speed of light c is the same for all inertial observers. This is the most important principle which underpins special relativity. It means that c^{\prime} = c, and so we can write

r^{\prime} = c t^{\prime}

which allows us to write Equations (1) and (2) as

x^{2} + y^{2} + z^{2} -c^{2}t^{2} = 0 \text{ in } S \text{ (3) }


x^{\prime}{^2} + y^{\prime}{^2} + z^{\prime}{^2} - c^{2}t^{\prime}{^2} = 0 \text{ in } S^{\prime} \text{ (4) }

Using the Galilean Transformations, which can be written as

we can substitute xfor x^{\prime} (=x - vt), y^{\prime}=y, z^{\prime}=z and t^{\prime} =t in Equ. (4) to give us

(x-vt)^{2} + y^{2} + z^{2} - c^{2}t^{2} = x^{2} - 2vxt +v^{2}t^{2} + y^{2} + z^{2} - c^{2}t^{2}

which is the same  as Equ. (3), except for the two extra terms -2vxt \text{ and } v^{2}t^{2}. This shows that Equations (3) and (4) are not equal under a Galilean transformation, even though the two equations should be equal (as both are equal to zero).

Let us now show that Equations (3) and (4) are equal using the Lorentz transformations. These can be written as

where \gamma is the Lorentz factor, and is defined as

\gamma = \frac{ 1 }{ \sqrt { \left( 1 - v^{2}/c^{2} \right) } }

We will substitute these expressions for x^{\prime},y^{\prime},z^{\prime} and t^{\prime} into Equ. (4). When we do this, we have

\left( \gamma (x-vt) \right)^{2} + y^{2} + z^{2} - c^{2} \left( \gamma (t - vx/c^{2}) \right)^{2}

Multiplying this out, and dropping the y^{2} and z^{2} terms, we get

\gamma^{2}(x^{2} -2vxt +v^{2}t^{2}) -c^{2}\gamma^{2}\left(t^{2} - \frac{2vxt}{c^{2}} + \frac{v^{2}x^{2}}{c^{4}}\right)

\gamma^{2}x^{2} -2\gamma^{2}vxt + \gamma^{2}v^{2}t^{2} - c^{2}\gamma^{2}t^{2} +2\gamma^{2}vxt - (\gamma^{2}v^{2}x^{2})/c^{2}

The two terms 2\gamma^{2} vxt disappear and, gathering terms in x^{2} and t^{2} together, we can write

\gamma^{2}x^{2} - \left( \frac{ (\gamma^{2}v^{2}) }{ c^{2} } \right) x^{2} + \gamma^{2}v^{2}t^{2} - \gamma^{2}c^{2}t^{2}

\gamma^{2}x^{2} (1 - v^{2}/c^{2}) + \gamma^{2}t^{2}(v^{2}-c^{2})

Remembering that \gamma^{2} = \frac{ 1 }{ ( 1 - v^{2}/c^{2} ) }, and changing the sign of the t^{2} term, we can write

\frac{ 1 }{ (1 - v^{2}/c^{2} ) } \cdot (1 - v^{2}/c^{2}) \cdot x^{2} - \frac{ 1 }{ (1 - v^{2}/c^{2} ) } \cdot (c^{2} - v^{2}) \cdot t^{2}

which is

x^{2} - \frac{ 1 }{ (1 - v^{2}/c^{2} ) } \cdot c^{2}(1 - v^{2}/c^{2}) \cdot t^{2}

which is

x^{2} - c^{2}t^{2}, exactly as in Equ. (3).

We have therefore shown that Equations (3) and (4) are equal if we use the Lorentz transformations.

Yesterday I introduced Paul Dirac, number 10 in “The Guardian’s” list of the 10 best physicists. I mentioned that his main contributions to physics were (i) predicting antimatter, which he did in 1928, and (ii) producing an equation (now called the Dirac equation) which describes the behaviour of a sub-atomic particle such as an electron travelling at close to the speed of light (a so-called relativistic theory). This equation was also published in 1928.

The Dirac Equation

In 1928 Dirac wrote a paper in which he published what we now call the Dirac Equation.

The equation now known as the Dirac Equation describes the behaviour of an electron when travelling close to the speed of light. The equation now known as the Dirac Equation describes the behaviour of an electron when travelling close to the speed of light.

This is a relativistic form of Schrödinger’s wave equation for an electron. The wave equation was published by Erwin Schrödinger two years earlier in 1926, and describes how the quantum state of a physical system changes with time.

The Schrödinger eqation

The time dependent Schrödinger equation which describes the motion of an electron The time dependent Schrödinger equation which describes the motion of an electron

The various terms in this equation need some explaining. Starting with the terms to the left of the equality, and going from left to right, we have i is the imaginary number, remember i = \sqrt{-1}. The next term \hbar is just Planck’s constant divided by two times pi, i.e. \hbar = h/2\pi. The next term \partial/\partial t \text{ } \psi(\vec{r},t) is the partial derivative with respect to time of the wave function \psi(\vec{r},t).

Now, moving to the right hand side of the equality, we have
m which is the mass of the particle, V is its potential energy, \nabla^{2} is the Laplacian. The Laplacian, \nabla^{2} \psi(\vec{r},t) is simply the divergence of the gradient of the wave function, \nabla \cdot \nabla \psi(\vec{r},t).

In plain language, what the Schrödinger equation means “total energy equals kinetic energy plus potential energy”, but the terms take unfamiliar forms for reasons explained below.

Today (January 30th) marks the 50th anniversary of the last time The Beatles played live together, in the infamous “rooftop” concert in 1969. Although they would go on to make one more studio album, Abbey Road in the summer of 1969; due to contractual and legal wranglings the rooftop concert, which was meant to be the conclusion to the movie they were shooting, would not come out until 1970 in the movie Let it Be.

It is also true to say that some of the songs on Abbey Road were performed “live” in the studio with very little overdubbing (as opposed to separate instrument parts being recorded separately as was done on e.g. Sgt. Pepper). But, the rooftop concert was the last time the greatest band in history were seen playing together, and has gone down in infamy. It has been copied by many, including the Irish band U2 who did a similar thing to record the video for their single “Where the Streets Have no Name” in 1987 in Los Angeles.

The Beatles were trying to think of a way to finish the movie that they had been shooting throughout January of 1969. They had discussed doing a live performance in all kinds of places; including on a boat, in the Roundhouse in London, and even in an amphitheatre in Greece. Finally, a few days before January 30th 1969, the idea of playing on the roof of their central-London offices was discussed. Whilst Paul and Ringo were in favour of this idea, and John was neutral, George was against it.

The decision to go ahead with playing on the roof was not made until the actual day. They took their equipment up onto the roof of their London offices at 3, Saville Row, and just start playing. No announcement was made, only The Beatles and their inner circle knew about the impromptu concert.

The concert consisted of the following songs :

  1. “Get Back” (take one)
  2. “Get Back” (take two)
  3. “Don’t Let Me Down” (take one)
  4. “I’ve Got a Feeling” (take one)
  5. “One After 909”
  6. “Dig a Pony”
  7. “I’ve Got a Feeling” (take two)
  8. “Don’t Let Me Down” (take two)
  9. “Get Back” (take three)

People in the streets below initially had no idea what the music (“noise”) coming from the top of the building was, but of course younger people knew the building was the Beatles’ offices. However, they would not have recognised any of the songs, as these were not to come out for many more months. After the third song “Don’t Let Me Down”, the Police were called and came to shut the concert down. The band managed nine songs (five different songs, with three takes of “Get Back”, two takes of “Don’t Let Me Down”, and two takes of “I’ve Got a Feeling”) before the Police stopped them. Ringo Starr later said that he wanted to be dragged away from his drums by the Police, but no such dramatic ending happened.

At the end of the set John said

I’d like to thank you on behalf of the group and ourselves, and I hope we’ve passed the audition.

You can read more about the rooftop concert here.

Here is a YouTube video of “Get Back” (which may get taken down at any moment)



and here is a video on the Daily Motion website of the whole rooftop concert (again, it may get taken down at any moment).



Enjoy watching the greatest band ever perform live for the very last time!

In late October 2017, astronomers announced the first ever discovery of an asteroid (or comet?) coming into our Solar System from another stellar system. The object was first spotted on 19 October by the University of Hawaii’s Pan-STARRS telescope, during its nightly search for near-earth objects. Based on its extreme orbit and its rapid speed, it was soon determined that the object has come into our Solar System from somewhere else, and this makes it the first ever asteroid/comet with an extra-solar origin to have been discovered. Originally given the designation A/2017 U1, the International Astronomical Union (IAU) have now renamed it 1I/2017 U1, with the I standing for “interstellar”. 


The object, given the designation A/2017 U1, was deemed to be extra-solar in origin from an analysis of its motion.

In addition to its strange trajectory, observations suggest that the object also has quite an unusual shape. It is very elongated, being ten times longer than it is wide. It is thought to be at least 400 metres long but only about 40 metres wide. This was determined by the rapid and dramatic changes in its brightness, which can only be explained by an elongated object tumbling rapidly.

The object has also been given the name Oumuamua (pronounced oh MOO-uh MOO-uh), although this is not its official name (yet).  This means “a messenger from afar arriving first” in Hawaiian. In other respects, it seems to be very much like asteroids found in our own Solar System, and is the confirmation of what astronomers have long suspected, that small objects which formed around other stars can end up wandering through space, not attached to any particular stellar system.

To read more about this fascinating object, follow this link.

A couple of weeks ago I was in New York City. Although I have been to NYC many times since I first visited it in October 1985, this was the first time I had spent the night in Manhattan.

On the evening of Saturday 17th November I booked a ticket to go up the Empire State Building to take night-time photos. The cost of a ticket to the 80th floor is US$32, and I’d say that it’s good value. But, what is not is the extra $20 to go up to the 102nd floor. I had forgotten from my last time that this is not worth the extra money. Hopefully I’ll remember next time.

I was on the observation deck of the Empire State (86th floor) for about 2 hours taking photos and videos. I’ll post some of them over the next few weeks. Then, at about 11:30pm I walked to Time Square. I stopped to get a coffee and warm up a bit so got to Time Square at about 12:30am. The place was heaving, hundreds of people were milling around and many of the shops were open.

At about 1:30am I caught the subway to Brooklyn as I wanted to get a photo of Manhattan with the Brooklyn Bridge in the foreground. I took my photos from the Brooklyn Bridge Park, then walked back to Manhattan over the Brooklyn Bridge.

By this time it was lashing it down with rain and I was soaked. The rain had seeped through my winter coat and my sweater and trousers were pretty much wet through. But, as I walked over the bridge I caught sight of the Statue of Liberty illuminated (it was about 3am by this time). So I decided to walk to Battery Park to take photos of her at night.

When I got to Battery Park and set up my camera I discovered that my camera battery was dead from all the long exposure photos I’d been taking. So I got my spare battery out of my backpack, only to find that it too was dead. So, I didn’t get any photos of Lady Liberty at night. It was now 4am.

I then walked back to my hotel which was in the Little Italy part of Manhattan. The rain was still lashing it down, and by now my phone was getting damp leading to Google maps misbehaving. The app kept on going haywire every minute or two, so I couldn’t use it to guide me back from Battery Park to Little Italy. Instead I just tried to figure it out, and it took me two hours!

I collapsed into my bed at 6am, having spent nearly 12 hours wandering around nighttime Manhattan taking photos.

When I was walking across the Brooklyn Bridge at about 3am this great Simon & Garfunkel song kept playing in my head.

The Only Living Boy in New York

This song appears on Simon & Garfunkel’s last album Bridge Over Troubled Water. Written, of course, by Paul Simon, the “Tom” in the lyrics refers to Art Garfunkel. When they were teenagers in Queens they released a single and briefly called themselves”Tom & Jerry”.

By 1970 Simon and Garfunkel were arguing and about to go their separate ways. Garfunkel decided to have a go at acting, he appears in the movie Catch 22. Simon is wishing him the best for his part in the movie Garfunkel is filming in Mexico.

As is usual with Paul Simon, the song’s lyrics are exquisite.

Tom, get your plane right on time
I know your part’ll go fine
Fly down to Mexico
Da-n-da-da-n-da-n-da-da a
And here I am
The only living boy in New York

I get the news I need on the weather report
Oh, I can gather all the news I need on the weather report
Hey, I’ve got nothing to do today but smile
Here I am
The only living boy in New York

Half of the time we’re gone but we don’t know where
And we don’t know where
Here I am
Half of the time we’re gone, but we don’t know where
And we don’t know where

Tom, get your plane right on time
I know that you’ve been eager to fly now
Hey, let your honesty shine, shine, shine
Like it shines on me
The only living boy in New York
The only living boy in New York

Here is a video of this beautiful song. Enjoy!

A couple of weeks ago this fascinating version of the periodic table of the elements was the NASA Astronomy Picture of the Day (APOD). Most people have seen the periodic table of the elements, it is shown on the wall of most high school chemistry classrooms. But, what is totally fascinating to me about this version is it shows the origin of each element.

It has been a long process of several decades to understand the origin of the elements. In fact, we have not totally finished understanding the processes yet. But, we do know the story for most elements. All the hydrogen in the Universe was formed in the big bang. This is true for nearly all the helium too. A small amount of the 25% or so of helium in the Universe has been created within stars through the conversion of hydrogen into helium. But, not much has been created this way because most of that helium is further converted to carbon.

The only other element to be formed in the big bang is lithium. About 20% of the lithium in the Universe was formed in the big bang, the rest has been formed since,

Together, hydrogen and helium comprise 99% of the elements in the Universe by number (not by mass).


Where Your Elements Came From – from the NASA Astronomy Picture Of the Day (APOD) 24 October 2017.

I have decided to use this fascinating table as the basis for a series of blogs over the next few weeks to explain each of the 6 processes in these six boxes

The Bob Dylan song about which I am going to blog about today is his 1973 song “Forever Young”, another one of his songs included on his Vevo channel. It appears on his 1974 album Planet Waves in two different versions, a slow and a fast version. It is the slow version which is in the video included here. As an aside, Planet Waves is the only studio album which Dylan released through Asylum Records. Apart from the live album Before The Flood which is his next album after Planet Waves, all his other albums have been with Columbia Records.

“Forever Young” was recorded by Dylan in November 1973. The slow version runs for 4m57s and is the 6th track on Planet Waves, the last track on the first side of the record. The fast version (which is a shorter track at 2m49s) is the 7th track on the album, the first track on the second side of the record. Dylan first performed “Forever Young” live in January 1974 and his most recent live performance of it was in November 2011. He has performed it live a remarkable 493 times as of my writing this.


“Forever Young” was recorded by Bob Dylan in November 1973 and appears on his 1974 album Planet Waves. There are two version of the song on the album, a slow version and a fast version.


You may be familiar with a 1988 Rod Stewart song by the same name. Confusingly, it is not a cover version in the traditional sense, but bears such a remarkable similarity to Dylan’s song in both melody and some of the lyrics that Stewart agreed to share his royalties with Dylan (presumably to avoid a lawsuit).


The inspiration for the song was Dylan’s eldest son Jesse who was born in 1966. Dylan wrote “Forever Young” as a lullaby to his young son, and over the years it has been covered by many artists.

May God bless and keep you always
May your wishes all come true
May you always do for others
And let others do for you
May you build a ladder to the stars
And climb on every rung
May you stay forever young
Forever young, forever young
May you stay forever young

May you grow up to be righteous
May you grow up to be true
May you always know the truth
And see the lights surrounding you
May you always be courageous
Stand upright and be strong
May you stay forever young
Forever young, forever young
May you stay forever young

May your hands always be busy
May your feet always be swift
May you have a strong foundation
When the winds of changes shift
May your heart always be joyful
May your song always be sung
May you stay forever young
Forever young, forever young
May you stay forever young

Here is the official Vevo video of this great song. Enjoy!

There are quite a few ways to derive Einstein’s famous equation E=mc^{2}. I am going to show you what I consider to be the simplest way.  Feel free to comment if you think you know of an easier way.

We will start off with the relationship between energy, force and distance. We can write

dE = F dx \text{ (1) }

Where dE is the change in energy, F is the force and dx is the distance through which the object moves under that force.  But, force can also be written as the rate of change of momentum,

F = \frac{dp}{dt}

Allowing us to re-write Equation (1) as

dE = \frac{dp}{dt}dx \rightarrow dE = dp \frac{dx}{dt} = vdp \text{ (2) }

Remember that momentum p is defined as

p =mv

In classical physics, mass is constant. But this is not the case in Special Relativity, where mass is a function of velocity (so-called relativistic mass).

m = \frac{ m_{0} }{ \sqrt{ ( 1 - v^{2}/c^{2} ) } } \text{ (3) }

where m_{0} is defined as the rest mass (the mass of an object as measured in a reference frame where it is stationary).

Assuming that both m \text{ and } v can change, we can therefore write

dp =mdv + vdm

This allows us to write Equ. (2) as

dE = vdp = v(mdv + vdm) = mvdv + v^{2}dm \text{ (4) }

Differentiating Equ. (3) with respect to velocity we get

\frac{dm}{dv} = \frac{d}{dv} \left( \frac{ m_{0} }{ \sqrt{ (1 - v^{2}/c^{2}) } } \right) = m_{0} \frac{d}{dv} (1 - v^{2}/c^{2})^{-1/2}

Using the chain rule to differentiate this, we have

\frac{dm}{dv} = m_{0} \cdot - \frac{1}{2} (1 - v^{2}/c^{2})^{-3/2} \cdot (-2v/c^{2}) = m_{0}  (v/c^{2}) \cdot (1 - v^{2}/c^{2})^{-3/2} \text{ (5) }

But, we can write

(1 - v^{2}/c^{2})^{-3/2} as (1-v^{2}/c^{2})^{-1/2} \cdot (1-v^{2}/c^{2})^{-1}

This allows us to write Equ. (5) as

\frac{dm}{dv} = m_{0}  (v/c^{2}) \cdot (1 - v^{2}/c^{2})^{-1} \cdot (1 - v^{2}/c^{2})^{-1/2}

From the definition of the relativistic mass in Equ. (3), we can rewrite this as

\frac{dm}{dv} = \frac{ m v }{ c^{2} }(1-v^{2}/c^{2})^{-1}

Which is

\frac{dm}{dv} = \frac{ m v }{ c^{2} } \left( \frac{c^{2}}{c^{2}} - \frac{ v^{2}}{c^{2} } \right)^{-1} = \frac{ m v }{ c^{2} } \left( \frac{c^{2}-v^{2}}{c^{2}} \right)^{-1}  = \frac{ m v }{ c^{2} } \left( \frac{c^{2}}{c^{2}-v^{2}}   \right)

\frac{dm}{dv} = \frac{ m v }{ (c^{2}-v^{2}) } \text{ (6) }

So we can write

c^{2}dm - v^{2}dm = mvdv

Substituting this expression for mvdv into Equ. (4) we have

dE = vdp = vd(mv) = mvdv + v^{2}dm = c^{2}dm - v^{2}dm + v^{2}dm


dE = c^{2} dm

Integrating this we get

\int_{E_{0}}^{E} dE = c^{2} \int_{m_{0}}^{m} dm


E - E_{0} = c^{2} ( m - m_{0} ) = mc^{2} - m_{0}c^{2}

E - E_{0} = mc^{2} - m_{0}c^{2}

This tells us that an object has rest mass energy E_{0} = m_{0}c^{2} and that its total energy is given by

\boxed{ E = mc^{2} }

where m is the relativistic mass.