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# Instantaneous Velocity

Definition: Instantaneous velocity is defined mathematically:

 v (2)
Example: Table 2.1 gives data on the position of a runner on a track at various times.

 t(s) x(m) 1.00 1.00 1.01 1.02 1.10 1.21 1.20 1.44 1.50 2.25 2.00 4.00 3.00 9.00

Find the runner's instantaneous velocity at t = 1.00 s. As a first estimate, find the average velocity for the total observed part of the run. We have,

 = = = 4 m/s. (3)

From the definition of instantaneous velocity Eq.(2.2), we can get a better approximation by taking a shorter time interval. The best approximation we can get from this data gives,

 = = = 2  m/s. (4)

We can interpret the instantaneous velocity graphically as follows. Recall that the average velocity is the slope of the line joining P and Q (from Figure 2.1). To get the instantaneous velocity we need to take t 0, or P Q. When P Q, the line joining P and Q approaches the tangent to the curve at P (or Q). Thus the slope of the tangent at P is the instantaneous velocity at P. Note that if the trajectory were a straight line, we would get v = , the same for all t .

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