Calculus-Based Kinematics Glossary

The relation a = v dv/dx, derived from a = dv/dt and v = dx/dt via the chain rule. Used for position-dependent acceleration.

An arbitrary constant C introduced with each indefinite integral, determined by boundary or initial conditions.

The signed change in position: integral of v(t) dt over a time interval. Can be positive, negative, or zero.

The total path length traveled: integral of |v(t)| dt. Always non-negative.

A known value of position, velocity, or acceleration at a specific time, used to determine constants of integration.

The derivative of velocity with respect to time: a(t) = dv/dt = d^2x/dt^2. Measured in m/s^2.

The derivative of position with respect to time: v(t) = dx/dt. Measured in m/s.

The derivative of acceleration: j(t) = da/dt = d^3x/dt^3. Measured in m/s^3.

Acceleration that varies with time, velocity, or position. Requires calculus (not kinematic equations) to solve.

Differentiation: d/dt(t^n) = nt^{n-1}. Integration: integral of t^n dt = t^{n+1}/(n+1) + C.

A differential equation that can be written as f(v) dv = g(t) dt, allowing each side to be integrated independently.

The constant velocity reached when the drag force equals the driving force (e.g., gravity). Found by setting a = 0 in the ODE.