Example 1.2, 3ed
\(\dfrac{d^2x}{dt^2} = g\)
\(\Rightarrow \dfrac{dx}{dt} = gt\)
\(\Rightarrow x(t) = \frac{1}{2}gt^2\)
\(x(0) = 0\)
\(x(\textrm{T}) = h\)
\(x = \frac{1}{2}gt^2\)
\(\Rightarrow t = \sqrt{\dfrac{2x}{g}}\)
\(\Rightarrow \dfrac{1}{t} = \sqrt{\dfrac{g}{2x}}\)
\(x(T) = \frac{1}{2}gT^2\)
\(\Rightarrow \frac{1}{2}gT^2 = h\)
\(\Rightarrow T^2 = \dfrac{2h}{g}\)
\(\Rightarrow T = \sqrt{\dfrac{2h}{g}}\)
\(\dfrac{dx}{dt} = gt\)
\(\Rightarrow \dfrac{dx}{gt} = dt\)
\(\dfrac{dt}{T} = \dfrac{dt}{\sqrt{\dfrac{2h}{g}}}\)
\(\Rightarrow \dfrac{dt}{T} = dt \sqrt{\dfrac{g}{2h}}\)
\(\Rightarrow \dfrac{dt}{T} = \dfrac{dx}{gt} \sqrt{\dfrac{g}{2h}}\)
\(\Rightarrow \dfrac{dt}{T} = \sqrt{\dfrac{g}{2h}} \left( \dfrac{1}{t} \right) \dfrac{dx}{g}\)
\(\Rightarrow \dfrac{dt}{T} = \sqrt{\dfrac{g}{2h}} \left( \sqrt{\dfrac{g}{2x}} \right) \dfrac{dx}{g}\)
\(\Rightarrow \dfrac{dt}{T} = \dfrac{1}{\sqrt{2}\sqrt{h}} \left( \dfrac{1}{\sqrt{2}\sqrt{x}} \right) dx\)
\(\Rightarrow \dfrac{dt}{T} = \dfrac{dx}{2\sqrt{hx}} \)
\(\textsf{P}(a \leq x \leq b) = {\displaystyle \int^b_a} \dfrac{1}{2\sqrt{hx}} \\; dx \qquad 0 < a \leq b \leq h\)
\({\displaystyle \int^h_0} \dfrac{1}{2\sqrt{hx}} \\; dx = {\displaystyle \int^h_0} \left( \dfrac{1}{2\sqrt{h}} \right) x^{-\frac{1}{2}} \\; dx \)
\( = \left. \left( \dfrac{1}{\sqrt{h}} \right) x^{\frac{1}{2}} \right|_0^h = 1 - 0 = 1\)
\(\langle x \rangle = {\displaystyle \int^h_0} x \dfrac{1}{2\sqrt{hx}} \\; dx\)
\(= {\displaystyle \int^h_0} x^{\frac{1}{2}} \dfrac{1}{2\sqrt{h}} \\; dx\)
\(= \left. x^{\frac{3}{2}} \dfrac{1}{3\sqrt{h}} \right|^h_0\)
\(= h^{\frac{3}{2}} \dfrac{1}{3\sqrt{h}} - 0\)
\(= \dfrac{h}{3}\)