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

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To derive the velocity function v(t) from the force function F(x) = sqrt(x) * sin(x^2) for a 1 kg block on a frictionless surface, one must first express acceleration a in terms of force using Newton's second law, a = F/m. The relationship between acceleration, velocity, and position can be established using the chain rule, leading to the equation a = v * (dv/dx). By integrating this equation with respect to position x, one can find the velocity as a function of position, v(x). Finally, to obtain v(t), the relationship between position and time must be established, often requiring integration of the velocity function. This process illustrates the general method for deriving velocity from a given force function in kinematics.
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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).
 
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