Instantaneous

Instantaneous describes a quantity evaluated at a single instant of time. In mathematics and physical sciences, it often contrasts with quantities defined over an interval, such as average speed or total energy over a span. An instantaneous quantity is typically defined as the limit of a rate or value as the time interval tends to zero.

Examples include instantaneous velocity, which is the derivative of position with respect to time: v(t) = ds/dt. The instantaneous speed is the magnitude |v(t)|. The instantaneous rate of change generalizes to any function f(t): f′(t) = lim_{h→0} [f(t+h) − f(t)]/h. In physics, instantaneous power is the time derivative of work, P(t) = dW/dt, and instantaneous angular velocity is ω(t) = dθ/dt, among others.

In applied contexts, instantaneous quantities require an exact moment of observation or a limit process, and are typically approximated in practice by using very small time intervals. In everyday language, instantaneous implies immediate occurrence or zero perceivable delay, though in physics and mathematics perfect simultaneity remains an idealization.