Question

please explain a=dv/dt = dv/dx = dx/dt = dv/dx

=adx=vdv

Answer 1

To explain:

Expression of acceleration of an object in terms of Velocity , Displacement, Time.

Solution:

Acceleration is defined as the instantaneous rate of change of velocity with respect to time.

$\boxed{\therefore \: a = \dfrac{dv}{dt} }$

Now , Velocity is also defined as the instantaneous rate of change of displacement with respect to time.

$= > \: a = \dfrac{d( \frac{dx}{dt}) }{dt}$

$\boxed{ = > \: a = \dfrac{ {d}^{2} x}{d {t}^{2} } }$

Hence , acceleration can also be expressed as second order differentiation of displacement function.

Performing chain rule to 1st equation:

$\therefore \: a = \dfrac{dv}{dt}$

$= > \: a = \dfrac{dv}{dx} \times \dfrac{dx}{dt}$

$= > \: a = \dfrac{dv}{dx} \times v$

$\boxed{ = > \: a = v\dfrac{dv}{dx} }$

Hence , acceleration can also be represented as the product of velocity and the velocity gradient.

Hope It Helps.