2 MOTION IN 1 DIMENSION 2.3 Velocity
Figure 3: Graph of displacement versus timebody is always approximately constant in the interval t to t + ∆t. Thus,
v = lim
∆x dx∆t (^0) ∆t =^ dt ,^ (2.3)^
where dx/dt represents the derivative
→
of x with respect to t. The above definition
is particularly useful if we can represent x(t) as an analytic function, because it
allows us to immediately evaluate the instantaneous velocity v(t) via the rules of
calculus. Thus, if x(t) is given by formula (2.1) then
v =
dx
= 1 + t − t^3. (2.4)
dt
Figure 4 shows the graph of v versus t obtained from the above expression. Note
that when v is positive the body is moving to the right (i.e., x is increasing in
time). Likewise, when v is negative the body is moving to the left (i.e., x is
decreasing in time). Finally, when v = 0 the body is instantaneously at rest.
The terms velocity and speed are often confused with one another. A velocity
can be either positive or negative, depending on the direction of motion. The
conventional definition of speed is that it is the magnitude of velocity (i.e., it is v
with the sign stripped off). It follows that a body can never possess a negative
speed.