Adding Velocity Vectors: Relative Motion
Define a coordinate system and determine
the vector components
Y
PT + TG = PG
T
PT = 2meters
P
.
θ
G
TG = 0.5meters
X
Table
φ
Let’s assume the displacements occurred
simultaneously within a 2 second period.
We then rewrite the above vector equations
in terms of velocity.
1
Velocity of the person wrt the table
Velocity of the table wrt the ground
Sum of the vectors gives the velocity
of the person wrt the ground
Ψ
Assume
2
Observe the three planes in the animation below. Each plane is heading south with a speed of 100 mi/hr. Each
plane flies amidst a wind which blows at 20 mi/hr.
In the first case, the plane encounters a tailwind (from behind) of 20 mi/hr. The combined effect of the tailwind and
the plane speed provide a resultant velocity of 120 mi/hr.
In the second case, the plane encounters a headwind (from the front) of 20 mi/hr. The combined effect of the
headwind and the plane speed provide a resultant velocity of 80 mi/hr.
In the third case, the plane encounters a crosswind (from the side) of 20 mi/hr. The combined effect of the
headwind and the plane speed provide a resultant velocity of 102 mi/hr (directed at an 11.3 degree angle east of
south).
These three resultant velocities can be determined using simple rules of vector addition.
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Relative Motion You are a pilot originating out of Pittsburg destined to Miami which is 900 miles directly South. At cruise altitude, the jet stream is forecasted to be toward the North‐East at 80mph. To remain on schedule, the flight time must be 2hrs. Determine the plane flight velocity vector in the air. 4
Relative Motion Y
Define a coordinate system with Pittsburg at the origin. Y‐axis is along the North direction and X‐
axis toward the East. The general Relative Motion equation Subscript indices
1 = plane
2 = air
3 = ground
X
We can then rewrite the equation as the following
5
As a pilot operating an airborne vehicle you have direct control only on the maneuverability
through the air medium
.
N
W
E
Pittsburg to Miami is 900
miles due South on the
ground and the distance
must be covered in 2 hrs
time therefore the ground
speed is 450mph and
based on our coordinate
system this speed must
be in the negative Y
direction. Therefore we
can write
NE
θ=45o
jet stream is forecasted to be toward the North‐East at 80mph. 6
Relative Motion
Airplane motion wrt ground
Air mass motion wrt ground
Airplane motion wrt air
2 equations and 2 unknowns X components:
Y components:
7
(Westerly)
(Southerly)
Direction = tan‐1 (‐506.7/‐56.7)=83.6o *
* Both components are negative,
therefore in 3rd quadrant Must add 180 degrees
Direction = 263.6o
N
Speed = SQRT(56.72+506.72)=509.9 mph
E
W
S
Aircraft Heading
8
Clicker Quiz
You are trying to cross a river that flows due south with a strong current. You start out in your motorboat on the east bank desiring to reach the west bank directly west from your starting point. You should roughly head your motorboat
A) due west.
B) due north.
C) due south.
D) in a southwesterly direction.
E) in a northwesterly direction.
W
E
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Chapter 4
The main idea of Chapter 4 is quite simple: Horizontal and Vertical Motions
are independent. That’s it.
For example, a ball thrown horizontally with a speed v continues to move
with the same speed v in the horizontal direction, even as it falls with an
increasing speed in the vertical direction.
Similarly, the fall time is the same, whether a ball is dropped from rest
straight down, or thrown horizontally.
Simply put, each motion continues as if the other motion were not present.
10
h
R
Constant acceleration in y
11
What is this shape??
12
Gravity shapes the trajectory of projectile.
It falls below the imaginary straight path as if it was dropped from a stationary point. 13
Monkey and the Hunter
http://www.youtube.com/watch?v=cxvsHNRXLjw
14
If the initial velocity is horizontal, then θ=0
therefore the x component of the initial velocity is equal to the full speed
In this case, the initial velocity in the y‐direction is zero. the equations of motion, with x0 = 0 and y0 = h:
15
Zero Launch Angle
h
y
since θ=0, then voy=0 and vox= vo
and based on our coordinate system we have
xo=0, yo =h
x
The time required to reach the water independent of vo !!
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