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Let $x$ be displacement as a function of time $t$ and some other physical quantity $k$ such that

$ x = f(t,k) $

Now,

1) Will the acceleration $a$ be $\frac{\partial^2 x}{\partial t^2}$ or $\frac{d^2 x}{dt^2}$ ?

2) Will both of the expressions yield the same result?

Thank You

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  • 2
    $\begingroup$ What do you exactly mean by some other physical quantity $k$? In classical mechanics, $x$ is to be treated simply as $x(t)$ because there is always a specific position of a particle at each value of time. And any other physical quantity is to be treated as $k(x(t),p(t),t)$ or $k(x(t),\dot{x}(t),t)$. $\endgroup$ – Dvij Mankad Feb 28 at 18:31
  • $\begingroup$ My use of $k$ was just to make $x$ an implicit function, I just wanted to know that will $a$ will be defined as the second partial derivative of $x$ wrt $t$ or second derivative of $x$ wrt $t$ .... $\endgroup$ – J Shelly Feb 28 at 19:23
  • $\begingroup$ What is your specific application of that? $\endgroup$ – flaudemus Feb 28 at 19:34
  • $\begingroup$ consider a situation where x is a function of time and some additional parameter which cannot be differentiated in terms of t, so just differentiating x wrt t will give us (dimensionally speaking) a term of acceleration and another term which will NOT have dimension of acceleration, so i just want to clarify that if we have a function of an implicit function of x, is it correct to use partial derivative ? $\endgroup$ – J Shelly Feb 28 at 19:45
  • $\begingroup$ Say in some problem the x coordinate depends as x = t+y² taking partial derivative wouldn't help here as we don't know time dependence of y $\endgroup$ – Aditya Garg Feb 28 at 19:50

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