Referring to Figure 6, is the instantaneous velocity at t = 0.500s (a) greater than, (b) less than, or (c) the same as the instantaneous velocity at t = 1.00s ?

 

C O N C E P T U A L  C H E C K  P O I N T 

I N S T A N T A N E O U S   V E L O C I T Y

Referring to Figure 6, is the instantaneous velocity at t = 0.500s  (a) greater than, (b) less than, or (c) the same as the instantaneous velocity at t = 1.00s ?

R E A S O N I N G  A N D   D I S C U S S I O N

From the x-versus-t graph in Figure 6 it is clear that the slope of a tangent line drawn at  t = 0.500s is greater than the slope of the tangent line at  t = 1.00 s . It follows that the particle’s velocity at 0.500 s is greater than its velocity at 1.00 s.

A N S W E R

(a) The instantaneous velocity is greater at t = 0.500 s

More explanation

When velocity is constant, the average velocity over any time interval is equal to the instantaneous velocity at any time.
In general, a particle’s velocity varies with time, and the x versus- t plot is not a straight line.

 An example is shown in Figure 6, with the corresponding numerical values of x and t given in Table 1.

In this case, what is the instantaneous velocity at, say, at t = 1.00s ? As a first approximation, let’s calculate the average velocity for the time interval from to t(i)= 0s  to  t(f) = 2 s . 

Note that this time interval is centered at t = 1.00s . From Table 1 we see that x(i) = 0 and  x(f) =27.4 m  , thus V(avg)= 13.7 m/s . The corresponding straight line connecting these two points is the lowest straight line in Figure 6.

The next three lines, in upward progression, refer to time intervals from 0.250s to 1.75 s, 0.500 s to 1.50 s, and 0.750 s to 1.25 s, respectively. 
The corresponding average velocities, given in Table 2, are 12.1 m/s, 10.9 m/s, and 10.2 m/s. Table 2 also gives results for even smaller time intervals. 
In particular, for the interval from 0.900 s to 1.10 s the average velocity is 10.0 m/s. Smaller intervals also give 10.0 m/s. Thus, we can conclude that the instantaneous velocity at t = 1.00 s is   v = 10.0 m/s.

The uppermost straight line in Figure 6 is the tangent line to the x-versus-t curve at the time t = 1.00 s ; that is, it is the line that touches the curve at just a single point. Its slope is 10.0 m/s.
 Clearly, the average-velocity lines have slopes that  approach the slope of the tangent line as the time intervals become smaller. 
This is an example of the following general result:
• The instantaneous velocity at a given time is equal to the slope of the tangent line at that point on an x-versus-t graph.
Thus, a visual inspection of an x-versus-t graph gives information not only about the location of a particle, but also about its velocity.

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