In elementary textbooks, one of the first examples of conservation of momentum is the 'recoiling gun'.
From the conservation of momentum, the momen- momentum = MV
tum of the recoiling gun is equal and opposite to the momentum of the bullet.
Mass is a measure of resistance to change of motion, or inertia. There is only 'one kind' of mass (It so happens that gravitational attraction is proportional to inertial mass.)
The speeds of the bullet and the gun recoil are inversely proportional to their masses.
For example, if the mass of the gun is 40 times the mass of the bullet, the initial speed of the bullet will be 40 times that of the recoil.
speed of the bullet mass of the gun recoil speed of the gun mass of the bullet
We can see that this formula will not work at very high velocities. For example, if we were to have a 'super-gun' recoiling with a speed of V = 0.5 c, the bullet would, according to the classical calculation, have a speed of 20 c — in breach of all the principles of special relativity!
Relativistic calculation for the imaginary super-gun:
A super-gun fires a bullet with speed 0.999 c and recoils with speed 0.5 c. Calculate the ratio of the mass of the gun to the mass of the bullet, and compare it to the ratio of the speed of the bullet to the recoil speed of the gun. Going back to Table 15.1 we find
Velocity v Gamma factor y
Inserting values for the velocities and the y factor into Equation (16.10),
_ V _
The ratio of the rest masses gun to bullet turns out to be about 40, and yet the ratio of their speeds is just less than 2.
16.4.2 Radioactive decay A relativistic 'gun and bullet'
Radioactive alpha decay presents us with real-life examples of 'gun and bullet' situations in which the decay of one unstable nucleus to a second nucleus and an alpha particle (He4), which then travel in opposite directions. The recoiling rifle is the second nucleus and the bullet is the alpha particle. (There are just two particles in the final state, after the decay has taken place — see Appendix 16.3.)
A typical example is the decay of radium:
88Ra226 ^ 86Rn222 + a
The alpha particle has an energy of 4.8 MeV, which corresponds to a speed of about 0.03 c, too small for any noticeable relativis-tic effects.
However, in beta decay the 'bullet' is an electron, which has a mass about 7000 times smaller than that of the alpha particle, and consequently much greater speed. The problem here is that the 'gun' is more like a shotgun, emitting not one 'bullet' but two. The electron is accompanied by a neutrino, which takes up some of the momentum. For example, in the beta decay of bismuth the reaction is
The energy and momentum of the decay products is shared between the electron and the neutrino in a fashion which is not predetermined. The maximum possible energy of the electron, the so-called end point energy, is 1.16 MeV, where the electron
is taking all of the available energy, and the neutrino gets none. For that special case, we again have a bullet + recoil gun situation in beta decay.
The kinematics calculation is given in Appendix 16.3, but some of the results are given below:
Speed of electron = 0.952 c = 2.856 x 105 km/s Speed of 83Po210 recoil = 8 x 10-6 c = 2.4 km/s
Ratio of rest masses of 'gun' to 'bullet' = 3.8 x 105 1 No longer Ratio of speeds 'bullet' to 'gun' = 1.19 x 105 J equal.
A historical interlude: Albert Einstein (1879-1955)
Albert Einstein was born in Ulm, Germany, in 1879. He was slow to talk, but once he started he spoke in complete sentences. Even at the age of four he showed remarkable curiosity about the world and its physical laws. For instance, he was fascinated by the magnetic compass. Here was a needle 'isolated and unreachable, totally enclosed, caught in the grip of an incredible urge to point north'. He later called this his 'first wonder'.
In his autobiography he writes: 'At the age of 121 experienced a second wonder. It was Euclid's proof that the intersections of the three altitudes of a triangle meet at a point. This, although by no means evident, could be proved with such certainty that any doubt appeared out of the question.'
As a pupil Einstein was neither particularly good nor bad. He wrote of himself 'My main weakness was a poor memory for both words and texts.' He sat in class looking bored, dreamy-eyed, and half-smiling, and this attitude was obviously not appreciated by those trying to teach him. He was actually asked to leave school when in fifth grade, his exasperated teacher stating: 'Your mere presence spoils the respect of the class for me.' His dislike of enforced teaching prompted him later to write: 'I believe that it would be possible to rob even a healthy beast of prey of its voraciousness if it were possible, with the aid of a whip, to force it to eat, even when
not hungry, especially if the food handed out under such coercion were selected accordingly.'
He was modest, friendly and unpretentious, with an awe of the beauty and logic of physical laws. He wrote: 'The most incomprehensible thing about the world is that it is comprehensible' At the same time he recognised the power of his own mind. When judging a scientific theory (his own or someone else's), he asked himself if he would have made the universe the same 'had he been God'!
The examination system for entrance to the prestigious Zurich Institute of Technology failed to uncover Einstein's potential. Mathematics and Physics presented no problems, but he failed Biology and Languages in 1895. He passed at his second attempt, in 1896. Even at the institute, he was not the ideal pupil. The normal curriculum bored him, so instead of going to lectures he spent his time in the Physics laboratory, or else reading advanced research papers. James Clerk Maxwell was one of his heroes; his predictions in electromagnetism were just the kind of logical reasoning which Einstein himself was to employ later in his theory of relativity.
Einstein was fortunate to have a good friend in Marcel Grossmann, (1878-1936) who was conscientious and took good notes. When the time came for examinations Einstein used these for cramming, and duly graduated in 1900 as a teacher of Mathematics and Physics, which gave the qualification for a temporary job in a secondary school. Grossmann remained Einstein's friend for life. They were to publish a number of papers together. It was Grossmann who supplied the mathematical expertise when Einstein was formulating the general theory of relativity in later years.
In 1902, Einstein obtained more stable employment at the Bern patent office, where he worked until 1909. He must have made a reasonable impression, as he was promoted in 1906 to 'Technical Expert, Second Class'. The job at the patent office could not have been too demanding, since he was able to do most of his original work on relativity in his spare time — and maybe also during office hours!
This was an exciting time in the history of physics. Max Planck had proposed the light quantum hypothesis in 1900. It took a couple of years to gain general acceptance, but when it did it was just the kind of thing to appeal to Einstein's sense of adventure. What Planck had just done was 'to wipe the slate clean of preconceived prejudices', to quote the words to be used by Einstein some years later concerning his ideas on space and time. Further milestones:
1903: Married Mileva Maric; 2 sons, born 1904 and 1910
1909: Associate Professor of Theoretical Physics, Zurich
1914: Professor at Kaiser Wilhelm Institute, Berlin
1919: Divorced from Mileva
1919: Married Elsa
1921: Nobel Prize
1932: Professorship at Princeton
Einstein's famous paper on the special theory of relativity was published in 1905 and was entitled 'On the Electrodynamics of Moving Bodies'. The title reflects his deep insight into Maxwell's theory of electromagnetism. Maxwell's work involved the motion of electric charges and the propagation of electromagnetic waves. Einstein realised that motion was relative, and that it was necessary to consider the description of the phenomena from different frames of reference.
Einstein received the 1921 Nobel Prize for Physics for 'the discovery of the law of the photoelectric effect'. It is remarkable that the great man wrote a number of papers during the first years of the century, any one of which could have been deserving of the prize. In his paper published in 1905, he extended Planck's concept of the quantum of energy to a concept of 'atoms of light'. It took many years for this concept to be accepted by the physics community. One may surmise that the Nobel committee were nervous about awarding him the prize for the theory of relativity since it was at that time not well established, and there was some controversy as to the originators of the principles of the special theory.
Einstein had a great sense of humour. Writing about the time when his visa application to the US was opposed by a group of women who claimed he was 'a communist at heart', he said: 'Never before had I experienced from the fair sex such energetic rejection, or if I had then certainly not from so many at once.'
In a letter from an unknown correspondent, Einstein was asked: 'Because of gravity a person is sometimes upright on the spherical earth, sometimes upside down, sometimes at left angles, sometimes at right angles. Would it be that while upside down, people do foolish things, like falling in love?' Einstein replied: 'Falling in love is not at all the most stupid thing that people do, and gravitation cannot be held responsible for it'
We also find in Einstein's writings serious thoughts about the universe and religion. In response to a question from a Sunday school in New York City, 'Do scientists pray?' he said: 'Everyone who is seriously involved in the pursuit of science becomes convinced that a Spirit is manifest in the Laws of the Universe, vastly superior to that of man, in the face of which we, with our modest prowess, must feel humble. In this way the pursuit of science leads to a religious feeling of a special sort, which is indeed quite different from the religiosity of someone more naive' In another letter to the National Conference of Christians and Jews in the US, he wrote: 'If the believers of the present day religions would earnestly try to think and act in the spirit of the founders of these religions, then no hostility on the basis of religion would exist among the followers of the different faiths'.
In 1932, Einstein accepted an arrangement at the University of Princeton by which he was to spend five months a year there and the remaining seven months in Berlin. Shortly afterwards the Nazis came to power, and what was intended only as a visit became a permanent post. He became a US citizen in 1940.
In a letter dated 2 August 1939, Einstein told President Franklin Roosevelt that experimental work by Fredene and Irene Joliot in France and Enrico Fermi and Leo Szilard in America had shown that it might be possible to set up a nuclear chain reaction in uranium. It was conceivable that the vast amounts of power released could be used for 'extremely powerful bombs of a new type'. He also mentioned that Germany had stopped the sale of uranium from Czechoslovakian mines which had just been taken over. Roosevelt replied in October that he would set up a board consisting of the head of the Bureau of Standards and chosen representatives from the Army and Navy to investigate the possibilities of Einstein's suggestion — a prompt reply but hardly one to indicate the extreme urgency of the situation.
Einstein wrote again, both in March and in April 1940, to stress that interest in uranium had intensified in Germany and that the Kaiser Wilhelm Institute had been taken over by the government to carry out highly secret research on uranium.
He was not to know that by then the US government had realised the potential of nuclear power. The Manhattan Project to develop an atomic bomb was about to be launched. Einstein, however, was considered a 'left-leaning political activist' and a 'security risk' and those who had been engaged on the project were forbidden to consult with him.
Einstein wrote a fourth and final letter, dated 25 March 1945. However, it failed to reach President Roosevelt before his death on 12 April 1945. Obviously, Einstein was not aware that the atomic bomb project was already well under way. The first test explosion was in fact carried out in New Mexico in July 1945.
When Einstein heard of the bombing of Nagasaki in August 1945, he was shattered. His equation E = mc2 had explained where nuclear energy came from, but not how to make a bomb. With great sadness he now saw that a terrible weapon of war had been constructed based on his discovery of the mind.
In the last decade of his life, Einstein dedicated himself to the cause of nuclear disarmament. 'The war is won'he said in December 1945, 'but the peace is not' One of the last acts of his life was to sign a manifesto urging all nations to give up nuclear weapons.
Albert Einstein died on April 18, 1955 at Princeton.
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