Galilei's theory stated that if in every initial time interval (0,T) a body falls by a distance F(t), then in the subsequent time intervals of the same duration (T,2T),(2T,3T),(3T,4T),...,(NT,(N+1)T), the body falls by distances according to the odd multiples 3F(T), 5F(T),7F(T),...,(2N+1)F(T). Galilei's theory also stated that the distance fallen is proportional to the square of the time elapse from rest. Riccioli's data support such theory. Problem 1 asks you investigate whether the two statements from Galilei's theory are equivalent.
problem 1: let F(KT) denote the total distance fallen from rest in the time interval (0,KT)
A. write an equation for the distance fallen in the interval (NT(N+1)T) confirming to example 1.
equation (1): F((N+1)T)-F(NT)=?
B. Verify that F(KT) = K^(2)F(T) for every
non-negative integer K solves your version of equation (1).
C. verify that F(KT) = K^(2)F(T) is the only solution of your version of equation (1).
D. Verify that if equation (1) holds for every T>0, then F(QT) = Q^(2)F(T) for every rational Q>=0
E. determine conditions under which F(t) = t^(2)F(1) for every real t>=0.
(if there's a problem reading the typed version please refer to the picture as it has the problems more organized)