If you rearrange Newton’s 2nd law you end up with a=f/m so how does the feather and hammer fall at the same rate when they have clearly different masses? The force is the same but with different mass then you have different acceleration by that logic.
Another thought the force of gravity also relies on the masses of the 2 objects and their distance so that would be different as well.
Anonymous
acceleration a = weight / inertial mass = m'*g/m"
where :
m' is the gravitational mass
m'' is the inertial mass
since m' and m'' have been found to be equal , the acceleration a is equal to g for every mass ( this is , of course, true in the vacuum where friction doesn't affect falling motion)
Jeff
The correct expression for Newton's second law is ΣF = ma, or the sum of all forces is equal to mass times acceleration. On the left hand side, you have two forces, gravitational force and drag force. Drag force depends on the speed of an object, the air density, the cross-sectional area of the object and the drag coefficient. A feather will have a much greater drag force relative to its weight, so its acceleration will decrease more quickly than that of a hammer. Rearranging the equation properly, you will have a = F_d/m - g, where F_d is drag force.
Jeffrey K
The force of gravity is NOT the same for the two objects. Remember Newton's Law of gravity:
F = GMm/r^2
The gravitational force depends on the mass of the object and the mass of the earth.
Put this in for F in the equation a=F/m and the mass of the object (little m) cancels out.
It is a very profound deep fact that the m in F=ma and the m in the gravity law are exactly the same thing. This is not a coincidence. It led Einstein to the general Relativity theory.
Amy
The force of gravity is proportional to mass.
This is more intuitive when you think about the larger object (e.g. the Earth) whose gravity is pulling on the smaller object. More mass -> stronger gravitational force.
But in fact gravity is proportional to the masses of both objects that are pulling on each other:
F = GMm/d^2
where G is a constant and d is distance between the two objects.
To calculate acceleration, the mass m of the small object cancels out.
a = F/m = GM/d^2
Hence the constant g = (G)(mass of Earth) / (radius of Earth)^2 is the acceleration caused by gravity of any object near the Earth's surface.
In contrast, air resistance depends on an object's area, not mass. Air resistance exerts about the same force on both a hammer and a feather, which mean it decelerates the feather more than the hammer.
ignoramus
The force is mg, so substituting in your equation you get
a = mg /m,
and the m's cancel to leave a = g, in other words the acceleration = gravitational acceleration, which is the same for any mass.