This story was about Earth scientists who were shown a film supposedly proving that Martian (?) scientists had perfected an anti gravity machine. After being convinced that it could be done, the Earth scientists built one. This may have been part of an Earth vs Mars book of short stories, probably written in the fifties.
Other than the part indicating Martians, this sounds a lot like the short story "Noise Level" by Raymond F. Jones.from 1952. In it the military assembled a group of scientists and shows them a film clip of a supposed working anti-gravity device. Unfortunately the inventor is dead and the notes destroyed. The scientists "knowing" anti-gravity works are able to duplicate the efforts. Later they find the military faked the film.
A Discussion of the story can be viewed here:
Link to listing of anthologies in which story appears. The overall link shows the anthologies.
- Noise Level [Martin Nagle], (nv) Astounding Dec 1952
◾Stories for Tomorrow, ed. William Sloane, Funk & Wagnalls 1954
◾Stories for Tomorrow, ed. William Sloane, Eyre & Spottiswoode 1955
◾Best SF 5, ed. Edmund Crispin, Faber and Faber 1963
◾Spectrum 5, ed. Kingsley Amis & Robert Conquest, Gollancz 1966
◾The Astounding-Analog Reader, Volume Two, ed. Harry Harrison & Brian W. Aldiss, Doubleday 1973
◾The Edward De Bono Science Fiction Collection, ed. George Hay, The Elmfield Press 1976
Note, reversing the process and entering the anthology name can also be used. If you do that, it will give you the listing of short stories contained within it.