1.14 Velocity  (Page 4/6)

$$\begin{array}{ll}\left|\mathbf{v}\right|=\frac{\mathrm{DC}}{\mathrm{AC}}& =\frac{dx}{dt}\end{array}$$

There is one important difference between average velocity and instantaneous velocity. The magnitude of average velocity $\left|{\mathbf{v}}_{\mathrm{avg}}\right|$ and average speed $v_{\mathrm{avg}}$ may not be equal, but magnitude of instantaneous velocity $\left|\mathbf{v}\right|$ is always equal to instantaneous speed $v$ .

We have discussed that magnitude of displacement and distance are different quantities. The magnitude of displacement is a measure of linear shortest distance, whereas distance is measure of actual path. As such, magnitude of average velocity $\left|{\mathbf{v}}_{\mathrm{avg}}\right|$ and average speed $v_{\mathrm{avg}}$ are not be equal. However, if the motion is along a straight line and without any change in direction (i.e unidirectional), then distance and displacement are equal and so magnitude of average velocity and average speed are equal. In the case of instantaneous velocity, the time interval is infinitesimally small for which displacement and distance are infinitesimally small. In such situation, both displacement and distance are same. Hence, magnitude of instantaneous velocity $\left|\mathbf{v}\right|$ is always equal to instantaneous speed $v$ .

Components of velocity

Velocity in a three dimensional space is defined as the ratio of displacement (change in position vector) and time. The object in motion undergoes a displacement, which has components in three mutually perpendicular directions in Cartesian coordinate system.

$$\begin{array}{l}\Delta \mathbf{r}=\Delta x\mathbf{i}+\Delta y\mathbf{j}+\Delta z\mathbf{k}\end{array}$$

It follows from the component form of displacement that a velocity in three dimensional coordinate space is the vector sum of component velocities in three mutually perpendicular directions. For a small time interval when Δ t → 0,

$$\begin{array}{l}\mathbf{v}=\frac{d\mathbf{r}}{dt}=\frac{dx}{dt}\mathbf{i}+\frac{dy}{dt}\mathbf{j}+\frac{dz}{dt}\mathbf{k}\\ \mathbf{v}=v_x\mathbf{i}+v_y\mathbf{j}+v_z\mathbf{k}\end{array}$$

For the sake of clarity, it must be understood that components of velocity is a conceptual construct for examining a physical situation. It is so because it is impossible for an object to have two velocities at a given time. If we have information about the variations of position along three mutually perpendicular directions, then we can find out component velocities along the axes leading to determination of resultant velocity. The resultant velocity is calculated using following relation :

$$\begin{array}{l}v=\left|\mathbf{v}\right|=\surd (v_x^2+v_y^2+v_z^2)\end{array}$$

The component of velocity is a powerful concept that makes it possible to treat a three or two dimensional motion as composition of component straight line motions. To illustrate the point, consider the case of two dimensional parabolic motions. Here, the velocity of the body is resolved in two mutually perpendicular directions; treating motion in each direction independently and then combining the component directional attributes by using rules of vector addition

Similarly, the concept of component velocity is useful when motion is constrained. We may take the case of the motion of the edge of a pole as shown in the figure here. The motion of the ends of the pole is constrained in one direction, whereas other component of velocity is zero.

The motion in space is determined by the component velocities in three mutually perpendicular directions. In two dimensional or planar motion, one of three components is zero. 10. The velocity of the object is determined by two relevant components of velocities in the plane. For example, motion in x and y direction yields :