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Describing motion using two dimensional vectors.
Typology: Slides
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Steps to add Vectors
Head to Tail method
set up a coordinate sys
draw a sketch of the vectors to be added
break up all vectors into x and y components
add all x components
add all y components
Position Vector in 2D
I am standing (5 miles, 7 miles). x and y coordinates
j
miles
i
miles
r
ˆ
7
ˆ
5
=
v
j
and
i
ˆ
ˆ
Write this as:
Unit vectors givedirection. They do not changemagnitude.
Any vector can bewritten in terms of theunit vectors:
To find the magnitude ofvector A:
To find components:
j
i
y
x
v
A
v
x
A
y
A
(
)
(
)
2
2
y
x
From Pythagorean Theorem
cos
x
θ
sin
y
θ
45
sin
1
1
r
r
y
=
2-D displacement Vector
1 vr
Find x and y components of r
1
, r
2
.
2 vr
All in x direction.
in x and ydirection.
y
r
1 v
x
r
1 v
x
x
x
r
r
R
2
1
=
2 vr
x
r
1 v
x
R
y
y
y
r
r
R
2
1
=
y
r
2 v
y
R
y
r
1 v
45
o
45
cos
1
1
r
r
x
=
2
2
r
r
x
=
2
2
r
r
x
=
x
x
x
r
r
R
2
1
=
2 vr
x
r
1 v
x
R
45
cos
1 r
2 r
y
y
y
r
r
R
2
1
=
y
r
2 v
y
R
y
r
1 v
0
45
sin
1
r
m
m
sin
= + = m 1.
12
=
m
m
m
cos
x
R
y
R
m
1 .
7
=
Instantaneous Velocity
dt
r
d
t r
v
t
inst
v
v
v
=
Δ Δ
=
→
Δ
0
lim
1 vr
2 v r^ vr
Δ
v r
Zoom in
1 vr
' 2 vr
v r
becomes
tangent topath as
t
shrinks to
zero
inst
v
v
is tangentto path atany point
General Equations for 2-D motion
j
y
i
x
r
ˆ
ˆ
=
v
j
v
i
v
v
y
x
ˆ
ˆ
=
v
{
}
{
}
j t a t v y i t a t v x r
y
y
x
x
ˆ
ˆ
2
1 2
1
1
2
1 2
1
1
{
}
{
}
j t a v i t a v v
y
y
x
x
ˆ
ˆ
1
1
=
v
Clicker Question Sequence: 2D motion 1 (ignore air friction in these questions)
A ball is thrown into the air at an angle of 30
o
from the horizontal.
Consider the motion of the ball after leaving the hand and beforehitting the ground.When the ball reaches the highest point in its trajectory describethe velocity of the ball.
The velocity of the ball...
The magnitude of the ...
The magnitude of the ...
None of the above.
50%
19%
31%
0%
The velocity of the ball is zero.
The magnitude of the velocity is atits greatest value.
The magnitude of the velocity is atits smallest value (but greater thatzero).
None of the above.
Initial
response(s)
Correct response
A projectile is launched from a cannon at three different angles: 30
o
, 60
o
and 90
o
.
In each case the projectile is launched with the
same speed.At which angle does the projectile spend the most time in the air?
30o
60o
90o
In all launches, the p...
Not enough information.
20% 20%
20%
20%
20%
Initial
response(s)
Correctresponse
30
o
60
o
90
o
In all launches, the projectilespends the same amount oftime in the air.
Not enough information.
A. Both are hit simultan...
B.
0%
43%
57%
A battleship simultaneously fires two shells with the same initialspeed at enemy ships. If the shells follow the parabolictrajectories shown, which ship gets hit first?
A.
Both are hit simultaneously.
B.
Initial
response(s)
Correctresponse
Developed by OSU
What equation should we use for the motion in the xdirection?
What equation should we use for the motion in the ydirection?
t
v
x
x
=
Δ
t
a
v
v
y
y
y
=
1
2
1 2
1
t
a
t
v
y
y
y
=
Δ
x
x
v
v
1
=
Accelerated motion
Constant velocity
a
y
= g = -9.
m
/
s
2
1
v
1
v
1x
g
v
1y
y
the
velocity decreases by 9.
m
s
each second
Δ
v
y
v
2
v
4x
v
4y
v
3
v
4
x
velocity is constant
y
velocity changes