Fun with dimensional analysis 3 – Vertical throw

A ball is thrown into the air with initial velocity \displaystyle v_0 and we wish to know how long it takes to return to the hand neglecting air resistance. The relevant physics is just the gravitational acceleration \displaystyle g. Thus we have

\displaystyle t \sim v_0^\alpha g^\beta \\ \Rightarrow \lbrack t \rbrack =\lbrack v_0 \rbrack^\alpha \lbrack g \rbrack^\beta \Rightarrow T = L^\alpha T^{-\alpha} L^\beta T^{-2\beta} \\ \Rightarrow 1 = -\alpha - 2\beta \mbox{ and } 0 = \alpha + \beta \\ \Rightarrow \beta = -1 \mbox{ and } \alpha = 1 \\ \therefore t \sim \frac{v_0}{g}

This result is fairly trivial to derive via differential equations (the missing constant is 2) but dimensional analysis is such an effective tool that even in this trivial cases I am wont to try it out.

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