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Acceleration

Show that the...

Question

Show that the rate of change of the speed is $\frac{d v}{d t} = \frac{(v_{x} a_{x} + v_{y} a_{y})}{\sqrt{v_{x}^{2} + v_{y}^{2}}} .$

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Solution

$v = \sqrt{v_{x}^{2} + v_{y}^{2}} \Rightarrow \frac{d v}{d t} = \frac{1}{2 \sqrt{v_{x}^{2} + v_{y}^{2}}} (\frac{d}{d t} v_{x}^{2} + \frac{d}{d t} v_{y}^{2}) = \frac{1}{2 \sqrt{v_{x}^{2} + v_{y}^{2}}} (2 v_{x} \frac{d}{d t} v_{x} + 2 v_{y} \frac{d}{d t} v_{y}) = \frac{v_{x} a_{x} + v_{y} a_{y}}{\sqrt{v_{x}^{2} + v_{y}^{2}}}$

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