Velocity

Both Fig. 3 and formula (11) effectively specify the location of the body whose motion we are studying as time progresses. Let us now consider how we can use this information to determine the body's instantaneous velocity as a function of time. The conventional definition of velocity is as follows:

Velocity is the rate of change of displacement with time.

This definition implies that

$$v = \frac{{\mit\Delta}x}{{\mit\Delta} t},$$ (12)

where $v$ is the body's velocity at time $t$, and ${\mit\Delta} x$ is the change in displacement of the body between times $t$ and $t+{\mit\Delta} t$.

How should we choose the time interval ${\mit\Delta} t$ appearing in Eq. (12)? Obviously, in the simple case in which the body is moving with constant velocity, we can make ${\mit\Delta} t$ as large or small as we like, and it will not affect the value of $v$. Suppose, however, that $v$ is constantly changing in time, as is generally the case. In this situation, ${\mit\Delta} t$ must be kept sufficiently small that the body's velocity does not change appreciably between times $t$ and $t+{\mit\Delta} t$. If ${\mit\Delta} t$ is made too large then formula (12) becomes invalid.

Suppose that we require a general expression for instantaneous velocity which is valid irrespective of how rapidly or slowly the body's velocity changes in time. We can achieve this goal by taking the limit of Eq. (12) as ${\mit\Delta} t$ approaches zero. This ensures that no matter how rapidly $v$ varies with time, the velocity of the body is always approximately constant in the interval $t$ to $t+{\mit\Delta} t$. Thus,

$$v = \lim_{{\mit\Delta}t\rightarrow 0}\frac{{\mit\Delta}x}{{\mit\Delta} t}= \frac{dx}{dt},$$ (13)

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 (11) then

$$v = \frac{dx}{dt} = 1 + t - t^3.$$ (14)

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.

Figure 4: Graph of instantaneous velocity versus time associated with the motion specified in Fig. 3

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.