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In a two dimensional motion of a body, p...

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

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`v=v_(x)hat(i)+v_(y)hat(j)`
Accleration `a=(dv_(x))/(dt)hat(i)+(dv_(y))/(dt)hat(j)`
Component of a along `v` will be, `(a.v)/(|v|)=(v_(x)(dv_(x))/(dt)+v_(y).(dv_(y))/(dt))/(sqrt(v_(x)^(2)+v_(y)^(y))`
Further, tangential acceleration of particle is rate of change of speed.
or `a_(t)=(dv)/(dt)=(d)/(dt)(sqrt(v_(x)^(2)+v_(y)^(2)))` or `a_(t)=(1)/(2sqrt(v_(x)^(2)+v_(y^(2))))[2v_(x).(dv_(x))/(dt)+2v_(y)(dv_(y))/(dt)]`
or `a_(t)=(v_(x).(dv_(x))/(dt)+v_(y).(dv_(y))/(dt))/(sqrt(v_(x)^(2)+v_(y)^(2)))`
Form Eqs. (i) and (ii), we can see that
or Tangential acceleration=component of acceleration along velocity.
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