# Physics 1A Introduction to Physics and Problem Solving

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```					      Lecture 4:
Vectors & Components
Questions of Yesterday
1) A skydiver jumps out of a hovering helicopter and a few
seconds later a second skydiver jumps out so they both
fall along the same vertical line relative to the helicopter.
1a) Does the difference in their velocities:
a) increase
b) decrease
c) stay the same
1b) What about the vertical distance between them?

2) I drop ball A and it hits the ground at t1. I throw ball B
horizontally (v0y = 0) and it hits the ground at t2. Which is
correct?
a) t1 < t2
b) t1 > t2
c) t1 = t2
Vector vs. Scalar Quantities
Vector Quantities: Magnitude and Direction
Ex. Displacement, Velocity, Acceleration

Scalar Quantities: Magnitude
Ex. Speed, Distance, Time, Mass

Vectors in 1
Dimension                             3        y (m)
Direction specified                       2
solely by + or -                         1
x (m)
x (m)     -3   -2 -1   1    2      3
-1
-3 -2 -1 0    1   2   3                     -2
-3
Vectors: Graphical Representation
Vector Quantities: Magnitude and Direction
Represent in 2D with arrow
Length of arrow = vector magnitude
Angle of arrow = vector direction

y (m)
R m at qo above x-axis
3
2                               Position of vector
1                                 not important
q              q        x (m)         Vectors of equal
-3 -2 -1           1q2   3           length & direction are equal
-1
-2
Can translate vectors for
q                       convenience (choose ref
-3                                    frame)
Must have same UNITS (true for scalars also)

A+B=?

A+B

A+B=B+A?

A+B                              B+A

YES!
A+B=B+A
order
Subtracting Vectors
A -> -A
Negative of vector = 180o rotation

A - B = A + (-
B)

A-B
Multiplying & Dividing Vectors by Scalars
2 * A = 2A             -2 * A = -2A

Ex. v = x/t
t=3s
Graphical Vector Techniques
N
1 box = 10 km
A plane flies from base
W            E
camp to lake A a distance
280 km at a direction 20o                  lake B           S
north of east. After
dropping off supplies, the
plane flies to lake B,
which is 190 km and
30.0o west of north from                           30o
lake A.
Graphically determine                                    lake A
the distance and
direction from lake B to
the base camp.
20o
base camp
Vector Components
Every vector can be described by its
components
Component = projection of vector on x- or y-
y                       y
axis

B                          R
q          y
x                          x
A                           R
x
From magnitude (R) and direction   Rx = Rcosq
(q) of R can determine Rx and Ry   Ry = Rsinq
Vector Components
Can determine any vector
from its components      y

R2 = Rx2 + Ry2
R = (Rx2 + Ry2)1/2                  R
q       y
tanq = Ry/Rx                         x
q = tan-1(Ry/Rx)               R
-90 < q < 90                  x
Vector Components
Can determine any vector          Careful!
from its components                    y

R2 = Rx2 + Ry2          (-x, +y)         (+x, +y)
R = (Rx2 + Ry2)1/2
II       I
x
tanq = Ry/Rx                 III       IV
q = tan-1(Ry/Rx)         (-x, -y)            (+x, -
-90 < q < 90                                  y)

I, IV: q = tan-1(Ry/Rx)
II, III: q = tan-1(Ry/Rx) + 180o
Important to know direction of vector!
Why are components useful?

When is magnitude of A + B = A + B ?

A           B

A+B
R x = A x + Bx +
Cx….
R = A + B + C…. =
R y = A y + By +
?
Cy….
q = tan-1(Ry/Rx)

lake B
Using components
determine the distance
and direction from lake
B to the base camp.                           30o

+
Rx = Ax + Bx lake A
Cx….
R y = A y + By +
20o     Cy….
base camp
q = tan-1(Ry/Rx)
-90 < q < 90
Vector Components: Problem #2
A man pushing a mop across a floor cause the mop to
undergo two displacements. The first has a magnitude of
150 cm and makes an angle of 120o with the positive x-
axis. The resultant displacement has a magnitude of 140
cm and is directed at an angle of 35.0o to the positive x-
axis. Find the magnitude and direction of the second
displacement.
Vector Components: Problem #3
An airplane starting from airport A flies 300 km east,
then 350 km at 30.0o west of north, and then 150 km
north to arrive finally at airport B. The next day,
another plane flies directly from A to B in a straight
line.
a) In what direction should the pilot travel in this direct
flight?
b) How far will the pilot travel in the flight?
Questions of the Day
1) Can a vector A have a component greater than its
magnitude A?
a) YES
b) NO

2) What are the signs of the x- and y-components
of A + B in this figure?
a) (x,y) = (+,+)
b) (+,-)
c) (-,+)
d) (-,-)

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 views: 3 posted: 8/15/2012 language: English pages: 17