Second Law with Changing Forces

Newton's second law, F = ma, always holds, but when F depends on time or position, a is not a fixed number anymore. It is a function. You cannot use the constant-acceleration equations from kinematics because they assume a never changes. Instead you go back to the definitions: a(t) = dv/dt, so velocity is the integral of acceleration over time. If F depends on x instead, you get acceleration directly at that location by plugging x into F(x) and dividing by mass, but getting velocity as a function of position takes a different trick involving v dv/dx = a, saved for later work with energy.

The mechanism for time-varying force is straightforward: write a(t) = F(t)/m, then integrate with respect to t to get v(t), adding the initial velocity as the constant of integration. If the object starts from rest, that constant is zero. For a force that depends on position, there is no time variable to integrate against yet, so you simply evaluate F at the given x and divide by m to get the instantaneous acceleration there. You are not finding a general function of time in that case, just a snapshot.

This topic is the bridge between the constant-force world of basic second law problems and the differential equations that show up with drag forces later in the unit, where force depends on velocity itself. It also sets up the variable-mass extension, where the system itself is losing material, like a rocket burning fuel, and the second law needs an extra term for the thrust from ejected mass. Each of these cases asks the same question: what does force depend on, and what tool from calculus turns that into motion?