Sometimes the average velocity is all you need to know about a particle's motion. For example, a race along a straight line is really a competition to see whose average velocity. Uav-x, has the greatest magnitude. The prize goes to the competitor who can travel the displacement ∆x from the start to the finish line in the shortest time interval, ∆t (Fig. 2.4). But the average velocity of a particle during a time interval can't tell us how fast, or in what direction, the particle was moving at any given time during the

interval. To do this we need to know the instantaneous velocity, or the velocity

at a specific instant of time or specific point along the path.

CAUTION How long is an instant? Note that the word "instant" has a somewhat differ-

ent definition in physics than in everyday language. You might use the phrase "It lasted

just an instant" to refer to something that lasted for a very short time interval. But in

physics an instant has no duration at all; it refers to a single value of time. 

To find the instantanecous velocity of the dragster in Fig. 2.1 at the point P1, we

move the second point P2 closer and closer to the first point P1 and compute the

average velocity Uav-x= ∆x/∆t over the ever-shorter displacement and time

interval. Both ∆x and ∆t become very small, but their ratio does not necessarily

become small. In the language of calculus, the limit of ∆x/∆t as ∆t approaches

zero is called the derivative of x with respect to and is written dxfdt. The

instamaneous velocity is the limit of the average velocity as the time interval

approaches zero; it equals the instantaneous rate of change of position with time.

We use the symbol Ux, with no "av" subscript, for the instantaneous velocity

along the x-axis, or the instantaneous -x-velocity: