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velocity, and acceleration of a falling body released from rest, and hence of Earths gravitational constant. Example 1 Galileis 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 {т, 27, 127, 3T]Ί3T, 4T], . . ., INT, (N + 1 , the body falls by distances according to the odd multiples 3F(T), 5F(T), 7F(T),., (2N+1)FT). Gallileis theory also stated that the distance fallen is proportional to the square of the time elapsed from rest. Ricciolis data support such theory [1], [2, p. 387]. (References are available on line from EWU accounts.) Problem 1 asks you to investigate whether the two statements from Gallileis 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 INT, (N + 1)T] conforming to example 1 (B) Verify that F(KT)- KF(T) for every non-negative integer K solves your version of equation (2) (C) Verify that F(KT) KF(T) is the only solution of your version of equation () (D) Verify that if equation (1) holds for every T 2 0, then F(QT) Q2F(T) for every rational Q2 0. (E) Determine conditions under which F() F(1) for every real 20.

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)

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