In a two dimensional motion of a body prove that tangential acceleration is nothing but component of acceleration along velocity.

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Solution

Let velocity of the particle be,
$\to v = v_{x}^i + v_{y}^j$
Acceleration $\to a = \frac{d v_{x}}{d t}^i + \frac{d v_{y}}{d t}^j$
Component of $\to a a l o n g \to v$ will be,
$\frac{\to a \cdot \to v}{| \to v |} = v_{x} \cdot \frac{d v_{x}}{d t} + v_{y} . \frac{d v_{y}}{d t}$ ...(i)
Further, tangential acceleration of particle is rate of change of speed.
or $a_{t} = \frac{d v}{d t} = \frac{d}{d t} (\sqrt{v_{x}^{2} + v_{y}^{2}})$
or $a_{t} = \frac{1}{2 \sqrt{v_{x}^{2} + v_{y}^{2}}} [2 v_{x} \cdot \frac{d v_{x}}{d t} + 2 v_{y} \frac{d b_{y}}{d t}]$
or $a_{t} = \frac{v_{x} \cdot \frac{d v_{x}}{d t} + v_{y} \cdot \frac{d v_{y}}{d t}}{\sqrt{v_{x}^{2} + v_{y}^{2}}}$
From Eqs. (i) and (ii), we can see that
$a_{t} = \frac{\to a \cdot \to v}{| \to v |}$
or Tangential acceleration= component of acceleration along velocity.

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