What relationship between distance traveled and time did galileo discover for freely falling objects released at rest?

Galileo Galilei was fondly called the "father of the observational astronomy". He was an Italian who mastered in many technical field of studies in order to bring about some of the most known principles, theories, laws, and equation both in physical sciences and astronomy. 

For free-falling objects (objects released from a certain height and was usually at rest), Galileo discovered that the distance varies with the square of time. In variation notation,
     d = kt²
where d is the distance, t is time, and k is the constant of proportionality. He also stressed that the motion of the free-falling body is independent of its weight.