The approximate age of ancient artifacts can be determined as long as the artifact is not older than 50,000 years. It's a technique called carbon-14 dating. All organisms that live on the Earth are based on carbon. This would be wood, plants, animals and of course humans.
There are three different isotopes of carbon, C-12, C-13, which are stable, and C-14. Carbon-14 is used for dating because it is radioactive and considered unstable. It will decay, or emit radiation, over a long period of time at a near consistent rate. The rate of decay, which can be measured, is 5,730 years equals one half-life.
Carbon-14, described by Argonne National Laboratory, is produced in Earth's atmosphere when N-14, Nitrogen, from New World Encyclopedia, atoms are bombarded by cosmic rays from space. They collide with the N-14 atoms and convert them into C-14 atoms. This collision changes a proton into a neutron. The C-14 then combines with oxygen to form carbon dioxide, CO-2. This is then absorbed by plants and animals.
All living things are replacing their carbon-14 as they continue living. Once they die the dating process begins. The radioactive carbon-14 begins to decay. It is very difficult to date organisms less than 1,000 years old because there is too much carbon-14 present. The same problem happens on the other end of the spectrum when an organism is older then 50,000 years, when there is not enough carbon-14 to measure.
Accelerated Mass Spectrometry (AMS) as explained by llnl.gov, was started by J.J. Thomson, from Nobel Prize.org, in 1897. "At first there were very few who believed in the existence of these bodies smaller than atoms. I was even told long afterwards by a distinguished physicist who had been present at my (1897) lecture at the Royal Institution that he thought I had been, 'pulling their legs." This by Scrips.edu.
Beginning in the early 1900's, mass spectrometry has been used to calculate the age of organisms using the carbon-14 isotope. Ions are accelerated in a field of millions of volts and smash through a gas or carbon foil. These ions then come to a stop in a gas ionization detector. Ion identity is determined by how fast they slow down.
Carbon-14 slows down faster than a nitrogen-14 isotope of the same mass. The AMS is a million times better at detecting the ratio of isotopes, by Colorado Edu, than by conventional means. It's this ratio that determines the age of the item tested. The AMS can recognize one carbon-14 isotope among a quadrillion different carbon atoms.
Dinosaur fossils, by All About Creation, cannot be age tested using the carbon-14 method because they are millions of years old. To determine their age the isotope's from uranium-238, uranium-235 and potassium-40 are used. They all have a half-life of more than a million years.
Dinosaur fossils are found in sedimentary rock. The isotope's needed to determine age are rarely found in sedimentary rock. Those isotopes are found in igneous rock made from cooled magma. No fossils are found in igneous rock because when it forms it would be too hot and destroy the bones. Scientists study neighboring layers of sedimentary rock that was formed at approximately the same time as the igneous rock. This gives them an approximate age of the fossil.
Using this method, scientists estimate the Earth is about 4.5 billion years old, USGS. Moon rocks are the same age lending weight to the theory that the Earth and Moon formed at the same time.