Scientists do not have direct measurements or observations of the initial moments of the Big Bang itself, as those conditions were vastly different from anything we can currently observe. However, they have used a combination of theoretical models, observational data, and scientific principles to infer the early moments of the universe and estimate its expansion rate.
The idea of the Big Bang theory emerged from observations such as the cosmic microwave background radiation (CMB) and the observed redshift of distant galaxies. The CMB is a faint, uniform radiation that permeates the universe and is considered to be the afterglow of the early hot and dense phase of the universe. By studying the properties of the CMB, scientists have gained valuable insights into the early universe.
The expansion rate of the universe, known as the Hubble constant, plays a key role in estimating the age of the universe and its expansion history. Scientists have measured the redshift of distant galaxies, which is a shift in the light they emit towards longer wavelengths. This redshift is caused by the expansion of space itself and is directly related to the velocity at which the galaxies are moving away from us.
By combining the observations of the redshift of galaxies with the theoretical framework of general relativity, scientists can estimate the expansion rate of the universe at different times in its history. By extrapolating this backwards in time, they can estimate the conditions and expansion rate of the universe at the time of the Big Bang.
It is important to note that the term "300,000,000 meters per second" mentioned in your question does not directly correspond to the speed of the initial explosion in the Big Bang. Rather, it is related to the approximate speed of light, which is the maximum speed at which information can travel in the universe according to our current understanding.
In summary, scientists have used a combination of theoretical models, observational data, and principles of physics to infer the early moments of the universe and estimate the expansion rate at the time of the Big Bang, but direct measurements of the initial moments are not possible.