Inertial and Gravitational Mass

The fact that, in the absence of air resistance, all projectiles have the same acceleration— the fact that the m’s canceled in Equation (1), has a deeper consequence than mere coincidence. In Newton’s second law, the m in the formula is the mass defined by the recoil definition of mass. Called inertial mass, it is the concept of mass that we get from the law of conservation of linear momentum.
............(1)

In Newton’s law of gravity, the projectile’s mass m in the formula F_g = Gmm_e/r_e² is what we should call the gravitational mass for it is defined by the gravitational interaction. It is the experimental observation that the m’s cancel, the observation that all projectiles have the same acceleration due to gravity, that tells us that the inertial mass is the same as gravitational mass. This equivalence of inertial and gravitational mass has been tested with extreme precision to one part in a billion by Etvös in 1922 and to even greater accuracy by R. H. Dicke in the 1960s.