The Infall Velocity Profile: A Jerk-Based Approach to Gravity in an Expanding Universe

We present a self-contained derivation of the infall velocity profile for a test particle moving under Newtonian gravity in an expanding cosmological background. Starting from a single differential relation between the squared acceleration and the velocity---expressed through the \emph{jerk}, defined as $j = \frac{1}{2}\,d(g^{2})/dv$---and employing only elementary calculus and the kinematic identity $g = \frac{1}{2}\,dv^{2}/dr$, we integrate twice to obtain a closed-form expression for $v(r)$. The boundary condition is set at the Hubble radius $R_{H} = c/H_{0}$, where the particle is momentarily at rest. The result, $v(r) \propto -[\ln(c/H_{0}r)]^{2/3}$, is derived in full detail, with every step motivated from first principles. The presentation is pitched at the level of an advanced high-school student or a first-year undergraduate, making the reasoning accessible to a broad audience.