Question

From a Brief History of Time, pg. 17:
"One can see why all the bodies fall at the same rate: a body of twice the weight will have twice the force of gravity pulling it down, but it will also have twice the mass. According to Newton's second law, these two effects will exactly cancel each other, so the acceleration will be the same in all cases." Does this mean that weight and mass cancel each other out? How does this happen?

Answer 1

it does not mean that weight and mass cancel each other, it implies that mathematically. The Second law (Newton's) states that force acting on a body of fixed mass is equal to its rate of change of momentum or simplified mathematically , \[Force = \frac{d(mv)}{dt}=m\times \frac{d(v)}{dt}=m \times a\]
when we compare two bodies falling vertically downward the force on each is their weight. \[m\times a_1 = W_1\] similarly \[2m\times a_2 = W_2\]Now consider the statement above " a body of twice the weight will have twice the force of gravity pulling it down" . This implies that \[W_2= 2\times W_1\]\[2m\times a_2 = 2(m\times a_1)\]which gives us \[a_1= a_2\] and we call this acceleration is due to gravity we denote it by 'g' and call it acceleration due to gravity.

Answer 2

The acceleration is the same in a uniform gravitational field.