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Instantaneous velocity

FIGURE 2.4

The concept of average velocity is inadequate for describing the details of a journey when the velocity is not constant. To obtain a better idea of how the velocity varies in time, we need to evaluate the ratio ∆ x /∆ t for many short time intervals. Is there any meaning in letting ∆ t get so small that it is essentially zero? Well, we know for example that a moving car has a definite velocity at any instant we choose to examine. The velocity at any instant of time, or point in space, is called the instantaneous velocity. Figure 2.4 shows what happens as the time interval is made shorter. As t 2 approaches t 1, the value of υ av changes. The chord cuts smaller and smaller sections of the curve, until finally it becomes the tangent to the curve at t 1. The instantaneous velocity at any instant is given by the slope of the tangent to the position-time graph at that time. Expressed mathematically,

(2.4)

The instantaneous velocity is the limiting value of the ratio ∆ x/∆t as ∆t approaches zero. Graphically, we have just seen how the slope of the chord approaches the slope of the tangent. Although the "real" ∆x and ∆t are vanishingly small, their ratio is the same for any pair of points on the tangent. So, for

FIGURE 2.5

accuracy in deter­mining υ graphically, we take large values of these quantities, as shown in Fig. 2.5. The process of finding the limiting value numerically is tedious. The tech­niques of calculus allow us to find this value for all functions of physical interest. In the notation of calculus. Eq.2.4 is written as

(2.5)

The instantaneous velocity is equal to the derivative of the position with respect to time. Physically, this means that υ is the rate of change of x with respect to t. The magnitude of the instantaneous velocity is the same as the instantaneous speed. From now on, the words speed and velocity, unless qualified, will mean the instantaneous values. Strictly speaking, we should write Eq.2.5 as υx = dx/dt. However, when dealing with one-dimensional motion the subscript is usually dropped for convenience.

 

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Displacement and velocity | Acceleration
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