ABSTRACT
Galileo’s results are fully consistent with Newton’s Laws. Imagine a block sliding without friction down a straight segment of track inclined at angle θ (Figure 9.2a). Ignoring friction and air drag, the only component of force in the direction of travel (parallel to the track) is Fgrav sin θ = mg sin θ. During a time interval Δt = tf − ti, the block moves between points i and f with constant acceleration a = g sin θ, or a = (vf − vi)/Δt where vi (or vf) is its initial (or final) speed. The distance s separating i and f is related to the average velocity by s v t v v tavg f i= = +∆ ∆‰( ) . Multiplying the expressions for a and s, we obtain
a s gs v v v v v vf i f i f i⋅ = = + − = −sin ( )( ) .θ
1 2
1 2
1 2
Since s sin θ = yi − yf, where yi (or yf) is the height of the initial (or final) point, the above equation may be rewritten as
1 2
1 2
2 2v v g y yf i i f− = −( ).
