Problem 7.13 Challenge

จาก \(v_{y}=\frac{dy}{dt}\)

\(v_{y}=\frac{dy}{dx}\frac{dx}{dt}\)

\(v_{y}= (2\alpha x + \beta )v_{x}; v_{x}=\frac{dx}{dt}\)

จาก \(a_{y}=\frac{dv_{y}}{dt}\)

\(a_{y}=\frac{d}{dt}((2\alpha x + \beta )v_{x})\)

\(a_{y}= (2\alpha x + \beta )\frac{dv_{x}}{dt}+ v_{x}\frac{d(2\alpha x + \beta )}{dt}\)

\(a_{y}= 2\alpha v_{x}^{2}; a_{x}=\frac{dv_{x}}{dt}= 0\)

\(v_{x}^{2} =\frac{a}{2\alpha}; a_{y}= a\)

จาก \(v = \sqrt{v_{x}^{2}+v_{y}^{2}}\)

\(v(x) =\sqrt{(1 + (2\alpha x + \beta )^{2})(\frac{a}{2\alpha})}\)

\(\therefore v(0) =\sqrt{(1 + (2\alpha (0) + \beta )^{2})(\frac{a}{2\alpha})}\)

\(v(0) =\sqrt{\frac{(1 + \beta ^{2})a}{2\alpha}}\;\blacksquare\)