What is $V$ in $a$=$V$$dv$/$dx$? [duplicate]

Question

$a$=instantaneous acceleration

$V$=instantaneous velocity

$x$=position

$dx$=small Chang in position

$a$=$dv$/$dt$

multiplying numerator and denominator by $dx$,we get

$a$=$dv$.$dx$/$dx$.$dt$

now we know $V$=$dx$/$dt$,

so,$a$ becomes $V$$dv$/$dx$

What is $V$ in $a$=$V$$dv$/$dx$??is it initial velocity?Or is it the velocity of the moment when we are finding the acceleration???is it velocity as a function of position or time???

Answer 1

Purely algebraically we see that $$\frac {\Delta v}{\Delta t}=\frac {\Delta v}{\Delta x}\times \frac {\Delta x}{\Delta t}.$$ It can be shown that in the limit as $\Delta t \rightarrow 0$ this becomes $$\frac {dv}{dt}=\frac {dv}{dx}\times \frac{dx}{dt}=\frac {dv}{dx}\times v.$$ $v$ is the instantaneous velocity.