Which experiment gave scientists reason to believe nuclear fission/fusion produced energy? Every piece of knowledge in science has a beginning lying in someone's experiment. I would like to know which experiment gave scientists the reason to believe nuclear fission/fusion existed and was instrumental in the development of the field of nuclear energy. 
I would also accept a thought experiment as an acceptable answer, as long as it answers the question.
 A: 
'Every piece of knowledge in science has a beginning lying in someone's experiment' 

A view that has long stopped being correct. In fact, I'd say the idea of nuclear energy was first established when Einstein formulated mass-energy equivalence: $E=m_0c^2$, which shows that there are enormous amounts of energy hidden in ordinary matter, if you can find a way to convert it...
A: The two answers here say the idea was conceived from einsteins famous equation, I would not have denied them had they also shown how people arrived from that equation to fission and fusion!
I also believe that "still" every idea is based on experiments, and I would say the idea of atleast fission was born with the idea of radioactive decay! 
Read here. 
Under history of radioactive decay they have explained that at first becquerel thought that some sort of radiation like x-rays were tampering with his experiments and producing the observations but many-many experiments later dons by various scientists it was established that different types of decays can occur which lead to different types of resultant nuclei.
As soon as you check the equations for these decays and the masses involved, you would know there is some mass missing! (Note radioactivity was discovered in 1896, before the famous equation by einstien ) Now "if" the scientists had the equation, it was easy to see there was loss of mass as energy. But we should note that people had even started doubting the energy-conservation principle at that time, only when new particles were predicted and found which balanced the equations things began to settle.
But any of these decay equations did not release sufficient energy which would have made it immediatelg evident that there was release of energy during the fission processes! The history section of nuclear fission clearly states that fission was discovered after 5 decades of work on radioactivity, at that time physicists had lots of experimental data and the einstein equation to figure out that a part of mass was being lost as energy. 
So, I would say experiments on radioactivity and the missing mass in equations for radioactive decay gave birth to the idea of energy generation of nuclear fission.
Now as far as fusion is concerned wikipedia's nuclear fusion  page clearly states that : 
Following the discovery of quantum tunneling by Friedrich Hund, in 1929 Robert Atkinson and Fritz Houtermans used the measured masses of light elements to predict that large amounts of energy could be released by fusing small nuclei. Building upon the nuclear transmutation experiments by Ernest Rutherford, carried out several years earlier, the laboratory fusion of hydrogen isotopes was first accomplished by Mark Oliphant in 1932.
You may give credit of discovery of fusion to either the experiments that led to idea of quantum tunneling, or the curiosity of robert atkinson and fitz houtermans for smashing light masses together. I prefer the latter, but thats just a choice I make.
PS: neutron which enables us to achieve fission was discovered in 1932, and fermi and his colleagues were experimenting bombarding uranium nucleus with neutrons and reported several fission reactions.  This is all mentioned in the nuclear fission page, and must have been a crucial stepping stone to discovery of fission
A: 
which experiment gave scientists the reason to believe nuclear fission/fussion existed 

Fusion was first. 
Francis William Aston built a mass spectrometer in 1919 and measured the masses of various isotopes, realizing that the mass of  helium-4  was less than 4 times that of hydrogen-1.
From this information, Arthur Eddington proposed hydrogen fusion as a possible energy source of stars.  
"Certain physical investigations in the past year, which I hope we may hear about at this meeting, make it probable to my mind that some portion of this sub-atomic energy is actually being set free in the stars. F. W. Aston's experiments seem to leave no room for doubt that all the elements are constituted out of hydrogen atoms bound together with negative electrons. The nucleus of the helium atom, for example, consists of 4 hydrogen atoms bound with 2 electrons. But Aston has further shown conclusively that the mass of the helium atom is less than the sum of the masses of the 4 hydrogen atoms which enter into it; and in this at any rate the chemists agree with him. There is a loss of mass in the synthesis amounting to about 1 part in 120, the atomic weight of hydrogen being 1.008 and that of helium just 4." Eddington 24 August 1920
At that time it was not understood that a neutron was distinct from a proton.  It was thought that the nucleus of helium 4 contained 4 protons and 2 electrons (instead of two protons and two neutrons), but Eddington's main idea that hydrogen fusing to helium released energy thereby powering stars was correct.
Eric Doolittle proposed a vague fission process in stars in 1919, but of course this was incorrect: "It seems very probable that when subjected to these inconceivably great temperatures and pressures, atoms may be broken up, and a part, at least, of their sub-atomic energy may be liberated. And it is only necessary to suppose that a part of the energy of the atom is in this way radiated into space in order that the life of a sun, or star, may be almost indefinitely prolonged".
Fission of heavy elements was discovered in the 1930s.  Enrico Fermi's experiments caused fission in 1934, be he did not realize that fission was occurring.  Otto Hahn and Fritz Strassmann concluded that upon neutron bombardment, uranium was broken into two lighter nuclei.  Lise Meitner and Otto Frisch made calculations concerning the large amount of energy released and introduced the term "fission".
A: According to the Wikipedia page on Nuclear fission the one experiment that really caused physicists to become excited about nuclear energy happened in 1938 when nuclear fission was discovered. The experiment involved splitting a Uranium atom, and the scientists noticed that Barium atoms were produced. The scientists concluded that the Uranium atom was like a drop of water that got split, and calculated that the amount of energy produced by the electric repulsion between the newly created Barium atoms was about 200 MeV. This result of 200 MeV matched what was predicted by $E=m{c}^{2}$ when comparing the mass of the original Uranium and the produced Barium.
It's interesting that the result of $E=m{c}^{2}$ was not necessary to determine that nuclear fission would produce large amounts of energy.
A: Einsteins mass-energy equivalence: $E=mc^2$ told scientists energy existed in mass which means in atoms and splitting an atom releases tremendous amounts of nuclear binding energy. Then they found shooting neutrons can split atoms and releases a lot of energy therefore knew they can start utilizing it. That's how they knew Nuclear fission AKA splitting atoms could be done.  
