vavg

vavg, often written as v̄ or v_avg, denotes the average velocity of an object over a given time interval. It is defined as the total displacement divided by the elapsed time. In one-dimensional motion, if an object's position is x(t), then v_avg over the interval t1 to t2 is v_avg = [x(t2) - x(t1)] / (t2 - t1). In vector form, v_avg = Δr / Δt, where Δr is the change in position.

If velocity is known as a function of time, v(t), the time-averaged velocity is v_avg = (1/(t2 - t1)) ∫_{t1}^{t2} v(t) dt. This agrees with the displacement formula when v is integrable and x(t) is differentiable.

v_avg is distinct from average speed. Average speed equals total distance traveled divided by the time interval and is always nonnegative, while v_avg can be zero or negative depending on the direction of displacement.

Examples: If a particle moves with constant velocity v0, then v_avg = v0. If x(t) = t^2, t1 = 0, t2 = 3, then v_avg = (x(3) - x(0))/3 = 3, while the instantaneous velocity at t = 3 is 6. If the object returns to its starting position, v_avg over that interval is zero despite nonzero instantaneous speeds.

Applications include kinematics analyses, experimental measurements of motion, and time-averaging of velocity fields in fluids or plasmas.