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Rectilinear kinematics describes particle motion along a straight line through position x(t), velocity v = dx/dt, and acceleration a = dv/dt. Three analysis cases arise: (1) constant acceleration — use the kinematic equations directly; (2) acceleration as a function of time, a(t) — integrate with respect to time; (3) acceleration as a function of position, a(x) — apply the chain rule identity a = v dv/dx to formulate a separable ODE. Selecting the correct method depends on how acceleration is specified.
Identify which case applies before choosing a solution method. Practice recognizing when to integrate a(t) and when to use a = v dv/dx. Always apply initial conditions after integrating.
Rectilinear kinematics answers a deceptively simple question: given how a particle accelerates, where is it and how fast is it moving at any time? The three kinematic quantities — position x(t), velocity v = dx/dt, and acceleration a = dv/dt — are connected by derivatives and integrals, and the solution strategy depends entirely on how the acceleration is specified.
Case 1: constant acceleration. If a is a fixed number, the familiar constant-acceleration kinematic equations apply directly: v = v₀ + at, x = x₀ + v₀t + ½at², and v² = v₀² + 2a(x − x₀). These are valid only when a truly does not change. Applying them to a variable-acceleration problem is the single most common error in introductory dynamics.
Case 2: acceleration as a function of time, a(t). Integrate once to get velocity: v(t) = ∫a(t) dt + C₁, and apply v(t₀) = v₀ to find C₁. Integrate again to get position: x(t) = ∫v(t) dt + C₂, with x(t₀) = x₀ determining C₂. Initial conditions are not optional — they are required to pin down the unique physical solution from the family of antiderivatives.
Case 3: acceleration as a function of position, a(x). Here you cannot integrate a dt because t does not appear explicitly in a. The solution is the chain rule identity: a = dv/dt = (dv/dx)(dx/dt) = v dv/dx. This transforms the equation into v dv = a(x) dx, a separable ODE that you integrate to find v as a function of x. Selecting the correct case before writing any equation is the key skill — it prevents the most common errors and immediately points to the right mathematical tool.
A final distinction worth internalizing: displacement is the net change in position (a signed number that can be zero even when the particle has moved far), while total distance traveled is the cumulative path length (always non-negative). A particle that travels 5 m right and 5 m back has zero displacement but 10 m of distance. Velocity changes sign when the particle reverses, so you must identify reversal points and split the integral to compute total distance correctly.