Motion Along A Straight Line
Key Ideas
โ The position x of a particle on an x axis locates the particle
with respect to the origin, or zero point, of the axis.
โ The position is either positive or negative, according
to which side of the origin the particle is on, or zero if the
particle is at the origin. The positive direction on an axis is
the direction of increasing positive numbers; the opposite
direction is the negative direction on the axis.
โ The displacement ฮx of a particle is the change in its
position:
ฮx = x2 โ x1.
โ Displacement is a vector quantity. It is positive if the
particle
has moved in the positive direction of the x axis and
negative if the particle has moved in the negative direction.
โ When a particle has moved from position x1 to position
x2 during a time interval ฮt = t2 โ t1, its average velocity
during that interval is
vavg = ฮx = x2 โ x1
ฮt t2 โ t1
โ The algebraic sign of vavg indicates the direction of
motion (vavg is a vector quantity). Average velocity does
not depend on the actual distance a particle moves, but
instead depends on its original and final positions.
โ On a graph of x versus t, the average velocity for a time
interval ฮt is the slope of the straight line connecting
the points on the curve that represent the two ends of
the interval.
โ The average speed savg of a particle during a time
interval
ฮt depends on the total distance the particle
moves in that time interval:
savg =total distance
ฮt
.
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Vectors
Key Ideas
โ Scalars, such as temperature, have magnitude only.
They are specified by a number with a unit (10ยฐC) and
obey the rules of arithmetic and ordinary algebra. Vectors,
such as displacement, have both magnitude and direction
(5 m, north) and obey the rules of vector algebra.
โ Two vectors a โ and b โ may be added geometrically by
drawing
them to a common scale and placing them head
to tail. The vector connecting the tail of the first to the
head of the second is the vector sum s โ. To subtract
b โ
from a โ, reverse the direction of b โ to get โb โ ; then add
โb โ to a โ. Vector addition is commutative and obeys the
associative law.
โ The (scalar) components ax and ay of any two-dimensional
vector a โ along the coordinate axes are found by dropping
perpendicular lines from the ends of a โ onto the coordinate
axes. The components are given by
ax = a cos ฮธ and ay = a sin ฮธ,
where ฮธ is the angle between the positive direction of
the x axis and the direction of a โ. The algebraic sign of a
component indicates its direction along the associated
axis. Given its components, we can find the magnitude
and orientation of the vector a โ with
a = โa2x + a2y
and tan ฮธ = ay
ax
.
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Motion in Two and Three Dimensions
Key Ideas
โ The location of a particle relative to the origin of
a coordinate system is given by a position vector r โ,
which in unit-vector notation is
r โ = xiห + yjห + zkห .
Here xiห, yjห, and zkห are the vector components of position
vector r โ, and x, y, and z are its scalar components (as
well as the coordinates of the particle).
โ A position vector is described either by a magnitudeand one or two angles for orientation, or by its vector or
scalar components.
โ If a particle moves so that its position vector changes
from r1 โ to r2โ , the particleโs displacement ฮr โ is
ฮr โ = r2 โ โ r1 โ.
The displacement can also be written as
ฮr โ = (x2 โ x1)iห + (y2 โ y1)jห + (z2 โ z1)kห
= ฮxiห + ฮyjห + ฮzkห .
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