Deriving v(t) from F(x), for Linear Motion

[silenteuphony]

TL;DR

Derive v(t) from F(x) = sqrt(x) * sin(x^2) for 1 kg block on level surface (or from F(x) in general if that function is too difficult)

Assuming I push a 1 kg block on a level surface, with no energy lost to friction, and I have an equation F(x) for force in terms of position, how would I derive an equation v(t) for velocity in terms of t? Specifically the function F(x) = sqrt(x) * sin(x^2), for 0 <= x <= sqrt(pi), if possible, but also just looking for the general procedure for any F(x).

 

[jedishrfu]

Search for “Using the chain rule in kinematics” or “velocity-dependent forces” to see how

$$a \;=\; \frac{dv}{dt}
\;=\; \frac{dv}{dx}\,\frac{dx}{dt}
\;=\; v\,\frac{dv}{dx}$$

is derived.

Here's a reference to check out:

https://farside.ph.utexas.edu/teaching/336k/Newton/node17.html