Transcript Slide 1

Vectors and Relative Motion
Vector Quantity
Fully described by both
magnitude (number
plus units) AND
direction
Represented by arrows
-velocity
-acceleration
-force
Scalar Quantity
Fully described by
magnitude (number
plus units) alone
-mass
-temperature
Adding Vectors
Vectors in one dimension are added
algebraically:
3 m, North
+ 4 m, North = 7 m, North
3m+4m=7m
3 m, North + 4 m, South = 1 m, South
3 m + (-4 m) = -1 m
For a vector-- Sign does not represent
value, it represents direction!
Traditionally: Up/Right (+)
Down/Left (-)
Adding Vectors in 2 Dimensions- Vectors add
Trigonometrically Using Head to Tail Method:
3.0 m + 4.0 m = 2.2 m
3.0 m + 4.0 m = 6.5 m
4.0 m
N
6.5 m
3.0 m
4.0 m
2.2 m
3.0 m
Vector diagrams show magnitude and
direction of vectors and their resultant!
8.0 N + 6.0 N = ? 2.0 N ≤ ? ≤ 14 N
Notice Vector Direction: In relation to + x axis
Vector Direction: By agreement, vectors are
generally described by how many degrees the
vector is rotated from the + x axis
30˚
30˚
150˚
Negative 2D vectors:
A
-A
180˚ opposite
Resolution (Decomposition) of Vectors
If you move a box 8.0 m @ 30.0˚ from O:
By Geometry:
8.0 m
30.0˚
4.0 m
6.9 m
The box has moved– 6.9 m to the right ( +x)
And
upward (+y)
These values would be the components of
the given vector !
d = 8.0 m
dy
Ø =30.0˚
dx
sinø =
dy
d
cosø =
dy = sinø(d) = sin30.0˚(8.0m)
= 4.0 m
Opposite Component!
dx
dx = cosø(d) = cos30.0˚(8.0m)
d
= 6.9 m
Adjacent Component!
V = √ Vx2 + Vy2
V
Vy
Θ
Vx
Adjacent component:
Opposite component:
tanΘ =
Vx = VcosΘ
Vy = VsinΘ
Vy
Θ = tan-1 (Vy/Vx)
Vx
Be careful of the
quandrant!
1) A man walks 5.0 km to the East and then
walks 3.0 km to the North. What is his
displacement from where he started?
2) What are the components of a vector
displacement of 12.0 m @ 32.0˚?
3) If a student walks 56.0 m North and then
turns West and walks another 85.0 m, what is
his displacement?
4) Vector B has components of dx = -22 m and
dy = - 33 m. What is the magnitude and
direction of this vector? What is the magnitude
and direction of – B ?
5) What is the resultant displacement when a
box is moved 5.00 m in the x direction and then
-7.50 m in the y direction?
6) What are the components of the vector
shown below?
A = 27.3 m
ø
Ø = 32.8˚
A
Adding Vectors Using Components
When adding two (or more) vectors, adding
the components will give the components of
the Resultant vector:
A golfer on a flat green putts a ball 7.50 m in
the Northeast direction, but misses the hole.
He then putts the ball 2.30 m @ 38.0˚ South
of straight East and sinks the putt for a
bogey. What single putt would have saved
par? d = 7.50 m @ 45.0˚
1
d2 = 2.30 m @ - 38.0˚
d1
d2
45.0˚
-38.0˚
Head to Tail—
On the head of
the first goes
the tail of the
next vector!
d1x = cos45.0˚(7.50 m) = 5.30 m
d1y = sin45.0˚(7.50 m) = 5.30 m
d2x = cos(-38.0˚)(2.30 m) = 1.81 m
d2y = sin(-38.0˚)(2.30 m) = -1.42 m
dx = d1x + d2x = 5.30 m + 1.81 m = 7.11 m
dy = d1y + d2y = 5.30 m + (- 1.42 m) = 3.88 m
d = √ dx2 + dy2 = √(7.11)2 + (3.88)2 = 8.10 m
Ø = tan-1 (dy / dx) = tan-1(3.88 / 7.11) = 28.6˚
d2
d1
The single
(resultant) putt:
d
ø
d = 8.10 m @ 28.6˚
1) What is the resulting displacement when an
object is moved 10.0 m to the North and then
5.0 m to the east?
2) A man leaves his house and walks 6.00 km
to the West and then turns and walks 3.50 km
to the South. What is his displacement?
3) A woman drives straight East for 65.0 km
and then turns 30.0˚ North of East and drives
another 33.0 km. What is her displacement?
4) A = 25.0 N @ 33.0˚ B = 57.7 N @ 152˚
Find the resultant when vector A is added to
vector B.
5) Add the following three vectors:
A = 225 m
α = 28.0˚
B
A
β
ø
α
C
B = 275 m
Β = 56.0˚
C = 325 m
ø = 15.0˚
Relative Velocity
Velocities are vectors and add like vectors:
A plane flies through the air at a speed of
255 m/s. The air speed is 33.0 m/s. The
velocity of the plane relative to the ground
depends upon direction:
In each case, the plane is heading (pointed in
that direction) South, but…
288 m/s
222 m/s 257 m/s @ 277˚
Remember: Default reference frame is Earth!
A boat travels at 12.0 m/s relative to the
water and heads East across a river that
flows North at 3.00 m/s. What is the speed
and direction of the boat relative to the
shore?
Vbw = 12.0 m/s @ 0˚ Vwg = 3.0 m/s @ 90.0˚
Vbg
ø
Vwg
Vbw
Vbg = (V12 + V22) = (12.02 + 3.002) = 12.4 m/s
Ø = tan-1(V2 / V1) = tan-1(3.00/12.0) = 14.0˚
Vbg = 12.4 m/s @ 14.0˚
1) A boat heads West across a stream that
flows South. What is the velocity of the boat
relative to the shore if it heads across with a
speed of 8.3 m/s while the water flows South
at 2.4 m/s?
2) An airplane heading due North at 325 m/s
encounters a wind of
m/s from the East.
What will the velocity of the plane be now?
3) A barge heading West down a still river
travels at 5.0 m/s. A man walks across the
barge from North to South at 2.0 m/s. What is
the velocity of the man as viewed from a
bridge above?
4) A boat wants to travel directly across a river
that flows South at 3.0 m/s. If the boat travels
at 7.0 m/s in still water, what heading must it
take to go straight across? With what speed
will the boat travel straight across?
5) An airplane has a velocity of 285 m/s @
215˚ while flying through a crosswind. What is
the heading of the plane? What is the velocity
of the wind?
6) A man in a blue car traveling at 25.0 m/s @
25.0˚ views a second red auto traveling at
32.0 m/s @ 215˚. What is the velocity of the
red car relative the the man in the blue car?
7) An airplane flies at 225 m/s @ 45.0˚ North
of east. A second plane flies at 175 m/s @
35.0˚ South of North. What is the velocity of
the the first plane relative to the second?
What is the velocity of the second plane
relative to the first?