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Direction of velocity
Average velocity may be different to instantaneous velocities in between the motion in either magnitude or direction or both. Consider the example of the tip of the second’s hand of a wall clock. It moves along a circular path of 2п r in 60 seconds. The magnitude of average velocity is zero in this period (60 seconds) as the second’s hand reaches the initial position. This is the overall picture. However, the tip of the second’s hand has actually traveled the path of 2п r, indicating that intermediate instantaneous velocities during the motion were not zero.
Also, the magnitude of average velocity may be entirely different than that of average speed. We know that distance is either greater than or equal to the magnitude of displacement. It follows then that average speed is either greater than or equal to the magnitude of average velocity. For the movement along semi-circle as shown in the figure below, the magnitude of velocity is 2r/30 m/s, where as average speed is п r/30 = 3.14 r/30 m/s. Clearly, the average speed is greater than the magnitude of average velocity.
Speed and velocity
Position – time plot is a convenient technique to interpret velocity of a motion. The limitation here is that we can plot position – time graph only for one and two dimensional motions. As a matter of fact, it is only one dimensional (linear or rectilinear) motion, which renders itself for convenient drawing.
On the plot, positions are plotted with appropriate sign against time. A positive value of position indicates that particle is lying on the positive side of the origin, whereas negative value of position indicates that the particle is lying on the opposite side of the origin. It must, therefore, be realized that a position – time plot may extend to two quadrants of a two dimensional coordinate system as the value of x can be negative.
On a position – time plot, the vertical intercept parallel to position axis is the measure of displacement, whereas horizontal intercept is the measure of time interval (See Figure). On the other hand, the slope of the chord is equal to the ratio of two intercepts and hence equal to the magnitude of average velocity.
Average velocity
Problem : A particle completes a motion in two parts. It covers a straight distance of 10 m in 1 s in the first part along the positive x - direction and 20 m in 5 s in the second part along negative x- direction (See Figure). Find average speed and velocity.
Motion in straight line
Characteristics of motion : One dimensional
Solution : In order to find the average speed, we need to find distance and time. Here total time is 1 + 5 = 6 s and total distance covered is 10 + 20 = 30 m. Hence,
The displacement is equal to the linear distance between initial and final positions. The initial and final positions are at a linear distance = -10 m. The value is taken as negative as final position falls on the opposite side of the origin. Hence,
The negative value indicates that the average velocity is directed in the opposite direction to that of the positive reference direction. We have discussed earlier that one dimensional motion consists of only two direction and as such an one dimensional velocity can be equivalently represented by scalar value with appropriate sign scheme. Though, the symbol for average velocity is shown to be like a scalar symbol (not bold), but its value represents direction as well (the direction is opposite to reference direction).
Also significantly, we may note that average speed is not equal to the magnitude of average velocity.
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