By https://physicsassistion.blogspot.com
Suppose a drag racer drives her AA-fuel dragster along a straight track (Fig. 2.1).
To study the dragster’s motion, we need a coordinate system. We choose the x-axis to lie along the dragster’s straight-line path, with the origin O at the starting
line. We also choose a point on the dragster, such as its front end, and represent
the entire dragster by that point. Hence we treat the dragster as a particle.
A useful way to describe the motion of the particle that represents the dragster
is in terms of the change in the particle’s coordinate x over a time interval. Suppose that 1.0 s after the start the front of the dragster is at point P1 19 m from the
origin, and 4.0 s after the start it is at point P2 277 m from the origin. The
displacement of the particle is a vector that points. Figure 2.1 shows that this vector points along the x-axis. The x-component of
the displacement is the change in the value of x, (277m — 19m) = 258 that took place during the time interval of (4.0s — 1.0s) = 3.0sWe
define the dragster’s average velocity during this time interval as a vector
quantity whose x-component is the change in x divided by the time interval:
(257m)/(3.0s) = 86m/s
In general, the average velocity depends on the particular time interval chosen. For a 3.0-s time interval before the start of the race, the average velocity
would be zero because the dragster would be at rest at the starting line and would
have zero displacement.
Let’s generalize the concept of average velocity. At time t1 the dragster is at
point P1 with coordinate x1, and at time t2 it is at point P2 with coordinate x2 .
The displacement of the dragster during the time interval from t¹ to t² is the vector from P1 to P2. The x-component of the displacement, denoted is the
change in the coordinate x:
∆x = x2-x1 (1.2)
The dragster moves along the x-axis only, so the y- and z-components of the displacement are equal to zero.
CAUTION : -
The meaning of ∆x Note that ∆x is not the ∆ product of and x; it is a single
symbol that means “the change in the quantity x.” We always use the Greek capital letter ∆ (delta) to represent a change in a quantity, equal to the final value of the quantity minus
the initial value—never the reverse. Likewise, the time interval from t1 to t2 is the
change in the quantity t: ∆t = t2-t1: (final time minus initial time).
The x-component of average velocity, or average x-velocity, is the x-component of displacement, ∆x, divided by the time interval ∆t during which
2.1 Positions of a dragster at two times during its run.
2.2 Positions of an official’s truck at
two times during its motion. The points p1 and p2 now indicate the positions of the
truck, and so are the reverse of Fig. 2.1.
the displacement occurs. We use the symbol Uav-x for average x-velocity (the
subscript “av” signifies average value and the subscript x indicates that this is the x-component):
Uav-x = x2-x1/t2-t1 = ∆x/∆t
As an example, for the dragster x1 = 19m, x2 = 277m, t1 = 1.0s, and t2 = 4.0s, so eq.(2.2) gives
Uax-x = 277m-19m/4.0s-1.0s = 258m/3.0s =86m/s
The average x-velocity of the dragster is positive. This means that during the time
interval, the coordinate x increased and the dragster moved in the positive
x-direction (to the right in Fig. 2.1).
If a particle moves in the negative x-direction during a time interval, its average velocity for that time interval is negative. For example, suppose an official’s
truck moves to the left along the track (Fig. 2.2). The truck is at x1 = 277m at t1 = 16.0s and is at x2 = 19m at t2 = 25.0 s. Then ∆x = (19m—277m) = -258m and ∆t = (25.0s—16.0s) =9.0s. The x-component of average
velocity is Uax-x = ∆x/∆t = (-258m)/(9.0s) = -29m/s. Table 2.1 lists
some simple rules for deciding whether the x-velocity is positive or negative.
CAUTION:-
Choice of the positive x-direction You might be tempted to conclude that
positive average x-velocity must mean motion to the right, as in Fig. 2.1, and that negative
average x-velocity must mean motion to the left, as in Fig. 2.2. But that’s correct only if
the positive x-direction is to the right, as we chose it to be in Figs. 2.1 and 2.2. Had we
chosen the positive x-direction to be to the left, with the origin at the finish line, the dragster would have negative average x-velocity and the official’s truck would have positive
average x-velocity. In most problems the direction of the coordinate axis will be yours to
choose. Once you’ve made your choice, you must take it into account when interpreting
the signs of and other quantities that describe motion!
With straight-line motion we sometimes call simply the displacement
and simply the average velocity. But be sure to remember that these are
really the x-components of vector quantities that, in this special case, have only
x-components.
Figure 2.3 is a graph of the dragster’s position as a function of time—that is,
an x-t graph. The curve in the figure does not represent the dragster’s path in
space; as Fig. 2.1 shows, the path is a straight line. Rather, the graph is a pictorial
way to represent how the dragster’s position changes with time. The points P1 and P2 on the graph correspond to the points P1 and P2 along the dragster’s path.
Line P1P2 is the hypotenuse of a right triangle with vertical side
∆x = x 2 - x 1
2.3 The position of a dragster as a
function of time.
and horizontal side ∆t = t1 -t2. The average x-velocity Uax-x = ∆x/∆t of the dragster equal the slope of the line P1P2—— that is, the ratio of the triangles vertical side ∆x to its horizontal side ∆t.
The average x-velocity depends only on the total displacement ∆x = x2-x1 that occurs during the time interval ∆t= t2-t1, not on the details of what
happens during the time interval. At time a motorcycle might have raced past
the dragster at point P1 in Fig. 2.1, then blown its engine and slowed down to
pass through point P2 at the same time t2 as the dragster. Both vehicles have the
same displacement during the same time interval and so have the same average
x-velocity.
If distance is given in meters and time in seconds, average velocity is measured in meters per second(m/s). Other common units of velocity are kilometers
per hour(Km/h), feet per second(ft/s), miles per hour(mi/h), and knots (1knot = 1 nautical mile /h = 6080 ft/h) Table 2.2 lists some typical velocity magnitudes.
Table 2.2 Typical Velocity
Magnitudes
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