Write on acceleration in terms of its component. (Or) Show that the acceleration is the second derivative of position vector with respect to time.

Explanation

In terms of components, we can write `veca=(dv_(z))/(dt)hati+(dv_(y))/(dt)hatj+(dv_(z))/(dt)hatk=(dvecv)/(dt)` Thus `a_(x)=(dv_(x))/(dt),a_(y)=(dv_(y))/(dt),a_(z)=(dv_(z))/(dt)` are the componets of instantaneous acceleration.
Since each component of velocity is the derivative of the corresponding coordinate, we can express the components `a_(x),a_(y)` and `a_(z)` as
`a_(x)=(d^(2)x)/(dt^(2)),a_(y)=(d^(2)y)/(dt^(2)),a_(z)=(d^(2)z)/(dt^(2))`
Then the acceleration vector `veca` itself is
`veca=(d^(2)x)/(dt^(2))hati+(d^(2)y)/(dt^(2))hatj+(d^(2)z)/(dt^(2))hatk=(d^(2)vecr)/(dt^(2))`
Thus acceleration is the second derivative of position vector with respect to time.