Griffiths Worked Examples

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$

$\langle x \rangle = {\displaystyle \int^h_0} x \dfrac{1}{2\sqrt{hx}} \; dx$

$= {\displaystyle \int^h_0} x^{\frac{1}{2}} \dfrac{1}{2\sqrt{h}} \; dx$

$= \left. x^{\frac{3}{2}} \dfrac{1}{3\sqrt{h}} \right|^h_0$

$= h^{\frac{3}{2}} \dfrac{1}{3\sqrt{h}} - 0$

$= \dfrac{h}{3}$