Transcript Physics 1A: Introduction to Physics and Problem Solving

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
What about 2 Dimensions?
Vectors in 1 Dimension
Direction specified
solely by + or -
y (m)
3
2
1
x (m)
x (m)
-3 -2
-1
0
1
2
3
-3
-2
-1
-1
-2
-3
1
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
Position of vector
not important
2
q
-3 -2
1
q
1q2
-1
-1
-2
-3
q
x (m)
3
Vectors of equal
length & direction are equal
Can translate vectors for
convenience (choose ref frame)
Adding Vectors: Head-to-Tail
Must have same UNITS (true for scalars also)
Must add magnitudes AND directions..how?
A+B=?
Head-to-Tail Method
A+B
Adding Vectors: Commutative Property
A+B=B+A?
A+B
B+A
YES!
A+B=B+A
Can add vectors in any order
Subtracting Vectors
A -> -A
Negative of vector = 180o rotation
A - B = A + (-B)
A-B
Multiplying & Dividing Vectors by Scalars
2 * A = 2A
Ex. v = x/t
t=3s
-2 * A = -2A
Graphical Vector Techniques
A plane flies from base
camp to lake A a distance
280 km at a direction 20o
north of east. After
dropping off supplies, the
plane flies to lake B, which
is 190 km and 30.0o west
of north from lake A.
N
1 box = 10 km
E
W
lake B
S
30o
Graphically determine the
distance and direction from
lake B to the base camp.
lake A
20o
base camp
Vector Components
Every vector can be described by its components
Component = projection of vector on x- or y-axis
y
y
B
A
Ry
x
From magnitude (R) and direction
(q) of R can determine Rx and Ry
q
x
Rx
Rx = Rcosq
Ry = Rsinq
Vector Components
Can determine any vector
from its components
y
R2 = Rx2 + Ry2
R = (Rx2 + Ry2)1/2
tanq = Ry/Rx
q = tan-1(Ry/Rx)
-90 < q < 90
Ry
q
x
Rx
Vector Components
Can determine any vector
from its components
R2 = Rx2 + Ry2
R = (Rx2 + Ry2)1/2
tanq = Ry/Rx
q = tan-1(Ry/Rx)
-90 < q < 90
Careful!
y
(-x, +y)
(+x, +y)
II
I
III
IV
(-x, -y)
x
(+x, -y)
I, IV: q = tan-1(Ry/Rx)
II, III: q = tan-1(Ry/Rx) + 180o
Important to know direction of vector!
Vector Addition: Components
Why are components useful?
When is magnitude of A + B = A + B ?
A
B
A+B
R = A + B + C…. = ?
Rx = Ax + Bx + Cx….
Ry = Ay + By + Cy….
q = tan-1(Ry/Rx)
-90 < q < 90
Vector Addition: Components
lake B
Using components
determine the distance
and direction from lake B
to the base camp.
30o
Rx = Ax + Bx + Cx….
lake A
Ry = Ay + By + Cy….
o = tan (R /R )
20q
y x
base camp -90 < q < 90
-1
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) (-,-)