Griffiths Worked Examples

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Example 1.2, 3ed

\(\dfrac{d^2x}{dt^2} = g\)

\(\Rightarrow \dfrac{dx}{dt} = gt\)

\(\Rightarrow x(t) = \frac{1}{2}gt^2\)


\(x(0) = 0\)

\(x(\textrm{T}) = h\)


\(x = \frac{1}{2}gt^2\)

\(\Rightarrow t = \sqrt{\dfrac{2x}{g}}\)

\(\Rightarrow \dfrac{1}{t} = \sqrt{\dfrac{g}{2x}}\)


\(x(T) = \frac{1}{2}gT^2\)

\(\Rightarrow \frac{1}{2}gT^2 = h\)

\(\Rightarrow T^2 = \dfrac{2h}{g}\)

\(\Rightarrow T = \sqrt{\dfrac{2h}{g}}\)


\(\dfrac{dx}{dt} = gt\)

\(\Rightarrow \dfrac{dx}{gt} = dt\)


\(\dfrac{dt}{T} = \dfrac{dt}{\sqrt{\dfrac{2h}{g}}}\)

\(\Rightarrow \dfrac{dt}{T} = dt \sqrt{\dfrac{g}{2h}}\)

\(\Rightarrow \dfrac{dt}{T} = \dfrac{dx}{gt} \sqrt{\dfrac{g}{2h}}\)

\(\Rightarrow \dfrac{dt}{T} = \sqrt{\dfrac{g}{2h}} \left( \dfrac{1}{t} \right) \dfrac{dx}{g}\)

\(\Rightarrow \dfrac{dt}{T} = \sqrt{\dfrac{g}{2h}} \left( \sqrt{\dfrac{g}{2x}} \right) \dfrac{dx}{g}\)

\(\Rightarrow \dfrac{dt}{T} = \dfrac{1}{\sqrt{2}\sqrt{h}} \left( \dfrac{1}{\sqrt{2}\sqrt{x}} \right) dx\)

\(\Rightarrow \dfrac{dt}{T} = \dfrac{dx}{2\sqrt{hx}} \)


\(\textsf{P}(a \leq x \leq b) = {\displaystyle \int^b_a} \dfrac{1}{2\sqrt{hx}} \\; dx \qquad 0 < a \leq b \leq h\)

\({\displaystyle \int^h_0} \dfrac{1}{2\sqrt{hx}} \\; dx = {\displaystyle \int^h_0} \left( \dfrac{1}{2\sqrt{h}} \right) x^{-\frac{1}{2}} \\; dx \)

\( = \left. \left( \dfrac{1}{\sqrt{h}} \right) x^{\frac{1}{2}} \right|_0^h = 1 - 0 = 1\)


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

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

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

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

\(= \dfrac{h}{3}\)