Instantaneous Velocity

Definition: Instantaneous velocity is defined mathematically:
 

$$\equiv \lim_{\Delta t \rightarrow 0}^{} {\frac{\Delta x}{\Delta t}}$$ (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,

$$\bar{v} {\frac{\Delta x}{\Delta t}} {\frac{9.00 {\:\rm m} - 1.00 {\:\rm m}}{3.00 {\:\rm s} -1.00 {\:\rm s} }}$$ (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,

$$\bar{v} {\frac{\Delta x}{\Delta t}} {\frac{1.02 \;{\:\rm m} - 1.00 \;{\:\rm m}}{1.01 \;{\:\rm s} -1.00 \;{\:\rm s}}}$$ (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 $\Delta$t $\rightarrow$ 0, or P $\rightarrow$ Q. When P $\rightarrow$ 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 = $\bar{v}$ , the same for all t .

Note:

• Instantaneous velocity gives more information than average velocity.
• The magnitude of the velocity (either average or instantaneous) is referred to as the speed.