2.3 INSTANTANEOUS VELOCITY
The average velocity of a particle during a time interval cannot tell us how fast, or in what direction, the particle was moving at any given time during the interval. To describe the motion in greater detail, we need to define the velocity at any specific instant of time or specific point along the path. Such a velocity is called instantaneous velocity, and it needs to be defined carefully.
Note that the word instant has a somewhat different definition in physics than it everyday language. You might use the phrase “It lasted just an instant” to refer to something that lasted for a very short time interval. But in physics an instant has no durati
at all; it refers to a single value of time.
To find the instantaneous velocity of the dragster in Fig. 2-1 at the point R, we imagine moving the second point ~ closer and closer to the first point ~. We compute the average velocity over these shorter and shorter displacements and time intervals. Both l:u and tM become very small, but their ratio does not necessarily become small. In the language of calculus the limit of l:u/tM as tM approaches zero is called derivative of x with respect to t and is written dx/dt. The instantaneous velocity is limit of the average velocity as the time interval approaches zero; it equals the rate of change of position with time. We use the symbol v, with no subscript, instantaneous velocity: u = lim 6x = dx (instantaneous velocity, straight-line motion).
We always assume that the time interval is positive so that v has the same sign as if the positive x-axis points to the right, as in Fig. 2-1, a positive v of v means that x is increasing and the motion is toward the right; a negative value of means that x is decreasing and the motion is toward the left. A body can have positive x and negative v, or the reverse; x tells us where the body is, while v tells us how moving. Instantaneous velocity, like average velocity, is a vector quantity. Equation (2- defines its x-component, which can be positive or negative. In straight-line motion.. other components of instantaneous velocity are zero, and in this case we will often v simply the instantaneous velocity. When we use the term “velocity,” we will all mean instantaneous rather than average velocity, unless we state otherwise.
The terms “velocity” and “speed” are used interchangeably in everyday lang but they have distinct definitions in physics. We use the term speed to denote dis traveled divided by time, on either an average or an instantaneous basis. Instantaneous speed measures how fast a particle is moving; instantaneous velocity measures how and in what direction it’s moving. For example, a particle with instantaneous velocity speed of 25 mls. Instantaneous speed is the magnitude of and so instantaneous speed can never be negative.
however, is not the magnitude of average velocity. When Matthew world record of 1986 by swimming 100.0 m in 48.74 s, his average “w.O m)/(48.74 s) = 2.052 mls. But because he swam two lengths in a 50-m and ended at the same point, giving him zero total displacement and zero for his effort! Both average speed and instantaneous speed are scalars, these quantities contain no information about direction.
2-5 (a) and (b) As the average velocity is calculated over shorter and shorter time intervals, its value approaches the instantaneous velocity. (c) In the limit t.t -+ 0 the slope of the line PJJ, approaches the slope of the tangent to the x-t curve at point PI’ The value of this slope equals instantaneous velocity v at point l’: in Fig. 2-1. We find the slope of the tangent by dividing
vertical interval (with distance units) along the tangent line by the corresponding horizontal interval (with time units). Here, v at l’: equals (160 m)/(4.0 s) = 40 mls.
FINDING VELOCITY ON AN x-t GRAPH
The velocity of a particle can also be found from the graph of the particle’s position a function of time. Suppose we want to find the velocity of the dragster in Fig. 2-1 point J]. As point Pz in Fig. 2-1 approaches point p., point P2 in the x-t graph of Fig approaches point PI’ This is shown in Figs. 2-5a and 2-5b, in which the average velocity is calculated over shorter time intervals &. In the limit that ~ t-+O, shown in 2-5c, the slope of the line p, P2 equals the slope of the line tangent to the curve at PI’ On a graph of position as a function of time for straight-line motion, the inst. velocity at any point is equal to the slope of the tangent to the curve at that If the tangent to slopes upward to the right, as in Fig. 2-5c, then its s is positive, the velocity is positive, and the motion is in the positive x-direction. If .
tangent slopes downward to the right, the slope and velocity are negative and the m is in the negative x-direction. When the tangent is horizontal, the slope is zero and . velocity is zero. Figure 2-6 illustrates these three possibilities. Note that Fig. 2-6 depicts the motion of a particle in two ways. Figure 2-6a is x-t graph, and Fig. 2-6b is an example of a motion diagram. A motion diagram the position of the particle at various times during the motion, like frames from a m or video of the particle’s motion, as well as to represent the particle’s velocity . each instant. Both x-t graphs and motion diagrams are helpful aids to understan
motion, and we will use both frequently in this chapter. You will find it worth your to draw both an x-t graph and a motion diagram as part of solving any problem involving motion.