## Average Velocity

The whole process can also be visualized in another way. Let t represent time, and suppose a particle is moving along the y-axis according to the equation y = t2. That is, at each time t the particle is at the point t2 on the y-axis. We then ask: what is meant by the velocity of the particle at time t0? Again we have the difficulty that the velocity is different at different times, and the calculus is needed to answer the question in a satisfactory way. Let us consider what happens to the particle between a time t0 and a later time t0 + Δt. The time elapsed is Δt, and the distance moved is Δy = 2t0 Δt + (Δt)2. If the velocity were constant during the entire interval of time, then it would just be the ratio Δy/Δt. However, the velocity is changing during the time interval. We shall call the ratio Δy/Δt of the distance moved to the time elapsed the "average velocity" for the interval;

The average velocity is not the same as the velocity at time t0 which we are after. As a matter of fact, for t0 > 0, the particle is speeding up; the velocity at time t0 will be somewhat less than the average velocity for the interval of time between t0 and t0 + Δt, and the velocity at time t0 + Δt will be somewhat greater than the average.

But for a very small increment of time Δt, the velocity will change very little, and the average velocity Δy/Δt will be close to the velocity at time t0. To get the velocity v0 at time t0, we neglect the small term Δt in the formula

vave = 2t0 + Δt,

and we are left with the value

v0 = 2t0.

When we plot y against t, the velocity is the same as the slope of the curve y = t2, and the average velocity is the same as the average slope.

Last Update: 2010-11-25