Transcript RESULTANT vector

Kinematic Equations
Δx = Vit + ½at2
Vf2 = Vi2 + 2aΔx
Vf = Vi + at
Vave = Δx/t
Vave = (Vi + Vf) / 2
a = (Vf – Vi) / t
R = (Vi2/g) sin(2θ)
Object Moving in the x Direction
Object Falling in the y Direction
Thus far we’ve been talking
about motion in the x
direction independently from
motion in the y-direction.
In the real world motion
occurs in a combination of
directions!
Imagine trying to swim directly across
a river that has a current flowing
perpendicular to your motion.
Push a light box with 50N of
force to the right and it moves to
the right.
50N
50N
Lift a light box with
50N of force
upwards and it
moves up.
The box moves
in this direction.
Lift and push a box at
the same time and it
will move to the right
and upward at the
same time.
50N
50N
The red arrows graphically
represent the magnitude
and direction of the applied
forces and are called
component VECTORS.
The black arrow is the
RESULTANT vector. It is a
combination of the two
other vectors.
50N
50N
You are looking down on an orange box on a blue floor and it is being pulled
to the east with a force of 50N while simultaneously being pulled North with
a force of 50N. Which direction does it move?
N

W  E

S
You are looking down on an orange box on a blue floor and it is being pulled
to the east with a force of 50N while simultaneously being pulled North with
a force of 50N. Which direction does it move?
Northeast
50N North
50N East
N

W  E

S
You are looking down on an orange box on a blue floor and it is being pulled
to the east with a force of 50N while simultaneously being pulled North with
a force of 25N. Which direction does it move?
N

W  E

S
You are looking down on an orange box on a blue floor and it is being pulled
to the east with a force of 50N while simultaneously being pulled North with
a force of 25N. Which direction does it move?
Because you are pulling
stronger in an eastward
direction the box will move
more towards the east than
the North.
25N North
50N East
N

W  E

S
“I go by Vector. It's a mathematical term, represented by an arrow
with both direction and magnitude. Vector! That's me, because I
commit crimes with both direction and magnitude. Oh yeah!” The
length of the arrow represents the magnitude and the direction
its pointed in represents the direction.
SCALAR
A scalar is a quantity with magnitude only.
Img via career pivot
Velocity, Force, Acceleration, Displacement, Momentum
Time, Mass, Volume, Temperature, Length, Speed
A person briskly walks across a 10m room and halfway back in
5 seconds. What is their average speed and average velocity?
Speed is a scalar and
uses total distance
travelled. It is not
concerned with direction.
Room is 10 meters long
Speed =
s=
s=
distance
time
Velocity =
d
t
15m
= 3m/s
5s
displacement
time
v=
v=
x
t
5m
= 1m/s
5s
Velocity is a vector an
uses displacement. It is
concerned with direction.
Displacement ≠ distance.
Displacement is how far
you are from your initial
position.
Determining the Resultant Vector of the two component vectors below:
Tail to Tip Method
1)
2)
3)
Start with any component vector.
Place the tail of any unused vector at the tip of it
Connect the tail of the first vector to the tip of the last vector.
RESULTANT!
50N
50N
50N
50N
• The Resultant is a simplification or combination of the other two vectors!
• Now you can do some trigonometry (SOHCAHTOA) and figure out magnitude
and direction of the resultant vector! (I know this makes you happy)
• Hopefully you realize that since the two sides are equal in
magnitude the angles must both be 45°.
• We can use the Pythagorean theorem (A2 + B2 = C2) to
determine the magnitude of the resultant or any of the Trig
Functions
RESULTANT!
71N
50N
50N
45°
50N
45°
50N
• The Resultant is a simplification or combination of the other two vectors!
• Now you can do some trigonometry (SOHCAHTOA) and figure out magnitude
and direction of the resultant vector! (I know this makes you happy)
Determining the Resultant
Tail to Tip Method
1)
2)
3)
Start with any component vector.
Place the tail of any unused vector at the tip of it
Connect the tail of the first vector to the tip of the last vector.
2m/s
2m/s
2.5m/s
2.5m/s
This is not a right triangle so more
information would be helpful.
Determining the Resultant Vector of the
two component vectors below:
Parallelogram Method
1)
2)
3)
The tail of both vectors originate at the same point
Draw in your parallelogram.
The resultant vector goes from the tail of both vectors
to the opposite side.
RESULTANT!
50N
50N
50N
50N
Determining the Resultant Vector of the
two component vectors below:
Parallelogram Method
1)
2)
3)
The tail of both vectors originate at the same point
Draw in your parallelogram.
The resultant vector goes from the tail of both vectors
to the opposite side.
2.5m/s
2m/s
2m/s
2.5m/s
Parallelogram
1)
2)
3)
All angles must add up to 360°.
Opposite Angles are Equal.
Opposite Sides are Equal
A
θ
θ=θ
θ
B
B
θ
θ
A
θ=θ
A=A
B=B
Use Law of Cosines to Calculate the Resultant
Graphical Method of Vector Resolution
1)
2)
3)
4)
Draw the vectors to scale with a ruler using tail to tip.
Draw in your resultant vector.
Measure its distance with a ruler and convert it back.
Use a protractor to determine the angle.
4.5in = 4.5m/s
4.5in
4m/s
4in
Check with trig!
2m/s
θ = Tan-1(4/2)
63.4°
2in
Vector Resolution Problems
For each problem draw the resultant using:
1. the tail to tip method
2. the parallelogram method
3. graphically using a ruler and protractor.
Determine the magnitude and direction using trigonometry.
[1] Find the resultant:
Vector 1: 9m east
Vector 2: 8m north
[2] Find the resultant:
Vector 1: 4m/s west
Vector 2: 5m/s south
[3] Find the resultant:
Vector 1: 10N west
Vector 2: 6N east
Vector 3: 7N north
A river is 100.0m wide and is flowing at rate of
2.0m/s downstream. You want to cross the river
at point A and can swim at a rate of 0.80 m/s.
a) Determine the magnitude and direction of
your motion with respect to the earth (get the
Resultant!).
b) How long does it take to cross the river?
c) How far downstream do you end up?
d) At what angle should you swim to end up
directly across from a?
0.8m/s
Current
100m
2m/s
A
Tail to Tip!
A river is 100.0m wide and is flowing at rate of
2.0m/s downstream. You want to cross the river
at point A and can swim at a rate of 0.80 m/s.
a) Determine the magnitude and direction of
your motion with respect to the earth (get
the Resultant!).
b) How long does it take to cross the river?
c) How far downstream do you end up?
d) At what angle should you swim to end up
directly across from a?
0.8m/s
Current
100m
2m/s
A
Resultant
2.2m/s
21.8°
0.8m/s
2m/s
Use Pythagorean theorem and/or SOHCAHTOA
to determine the magnitude and direction of
the resultant vector.
• Tan-1 (O/A) = Tan-1 (0.8/2) =21.8°
• Sin(21.8°) = 0.8/H thus H = 2.2m/s
The current is pushing you downstream due
A river is 100.0m wide and is flowing at rate of
East whereas you are swimming due north.
2.0m/s downstream. You want to cross the river Your speed across the river is unaffected by
the current since it has no impact on the
at point A and can swim at a rate of 0.80 m/s.
North South Direction. You are crossing the
river at a speed of 0.8m/s and it has it has a
a) Determine the magnitude and direction of
across of 100m. This is a simple
your motion with respect to the earth (get the distance
speed problem.
Resultant!). 2.2m/s @ 21.8°
S = d/t so 0.8m/s = 100m/t
b) How long does it take to cross the river?
c) How far downstream do you end up?
t = 125s
d) At what angle should you swim to end up
directly across from a?
0.8m/s
Current
100m
2m/s
A
A river is 100.0m wide and is flowing at rate of
2.0m/s downstream. You want to cross the river
at point A and can swim at a rate of 0.80 m/s.
a) Determine the magnitude and direction of
your motion with respect to the earth (get the
Resultant!). 2.2m/s @ 21.8°
b) How long does it take to cross the river?
t = 125s
c) How far downstream do you end up?
d) At what angle should you swim to end up
directly across from a?
0.8m/s
Current
100m
2m/s
A
Since we now know that your total time in
the water was 125s and that that your rate
downstream was 2m/s we can very easily
determine how far down stream you ended
up.
S = d/t and d = st
2m/s x 125s = 250m
So the river pushes you
downstream 250m in 125s.
A river is 100.0m wide and is flowing at rate of
2.0m/s downstream. You want to cross the river
at point A and can swim at a rate of 0.80 m/s.
a) Determine the magnitude and direction of
your motion with respect to the earth (get the
Resultant!). 2.2m/s @ 21.8°
b) How long does it take to cross the river?
t = 125s
c) How far downstream do you end up? d= 250m
d) At what angle should you swim to end up
directly across from a?
Don’t misunderstand a resultant.
Your South to North component of
motion (0.8ms) and your West to East
component of motion (2m/s) never
changes. The resultant is a combination
of the two. Its simply stating that the
addition of both of those vectors
results in a direction of motion that
looks like the resultant.But you can still
treat that motion individually as we just
did..
2.2m/s
100m
0.8m/s
2m/s
A
A river is 100.0m wide and is flowing at rate of
2.0m/s downstream. You want to cross the river
at point A and can swim at a rate of 0.80 m/s.
a) Determine the magnitude and direction of
your motion with respect to the earth (get the
Resultant!). 2.2m/s @ 21.8°
b) How long does it take to cross the river?
t = 125s
c) How far downstream do you end up? d= 250m
d) At what angle should you swim to end up
directly across from a?
Current
100m
0.8m/s
2m/s
A
A river is 100.0m wide and is flowing at rate of
2.0m/s downstream. You want to cross the river at
point A and can swim at a rate of 0.80 m/s. How far
downstream do you end up and how long does it
take you cross the river?
a) Determine the magnitude and direction of your
motion with respect to the earth (get the
Resultant!). 2.2m/s @ 21.8°
b) How long does it take to cross the river? t= 125s
c) How far downstream do you end up? 250m
d) At what angle should you swim to end up
directly across from a? Impossible
This is a trick question. You can
NEVER reach the point directly
across from A.
The current is moving faster
downstream than you can swim so
even if you swam directly upstream
the current would still win.
The 0.8m/s upstream would cancel
with some of the 2m/s and result in
a vector of 1.2m/s downstream.
Resultant is 1.2m/s
100m
0.8m/s
A
2m/s
Breaking a Vector into Components.
A golf ball is hit eastward at an angle of 30° with
respect to the ground with an initial velocity of
30.0m/s. What are are the horizontal and vertical
components of the golf ball’s motion?
Use Sin to get y-component
O
Sin(q ) =
H
Use Cos to get x-component
15.0m/s
Vy
30°
26.0m/s
Vx
A
Cos(q ) =
H
Breaking a Vector into Components.
A golf ball is hit eastward at an angle of 30° with
respect to the ground with an initial velocity of
30.0m/s. What are are the horizontal and vertical
components of the golf ball’s motion?
Now that you know the horizontal
and vertical components of motion
you could answer questions such
as:
•
•
•
15.0m/s
Vy
30°
26.0m/s
Vx
How long will the ball be in the air?
How high with ball get?
How far away from the spot it was
hit will it land?
Breaking a Vector into Components.
A golf ball is hit eastward at an angle of 30° with
respect to the ground with an initial velocity of
30.0m/s. What are are the horizontal and vertical
components of the golf ball’s motion?
Now that you know the horizontal
and vertical components of motion
you could answer questions such
as:
•
•
•
15.0m/s
Vy
30°
How long will the ball be in the air?
How high with ball get?
How far away from the spot it was
hit will it land?
The ball is going upwards initially at 15m/s and will rise
until gravity slows It to a complete stop at its highest
point. It will then fall and increase its velocity until it
strikes the ground with the same magnitude it was
launched with but in the opposite direction. The only
acceleration in the y direction is g (-10m/s2) and we
know initial and final velocity so we can figure out the
time it will take the ball to land using
Vf = Vi +at
26.0m/s
Vx
-15m/s = 15m/s -10m/s2 t
t = 3s
Breaking a Vector into Components.
A golf ball is hit eastward at an angle of 30° with
respect to the ground with an initial velocity of
30.0m/s. What are are the horizontal and vertical
components of the golf ball’s motion?
Now that you know the horizontal
and vertical components of motion
you could answer questions such
as:
•
•
•
15.0m/s
Vy
How long will the ball be in the air?
How high with ball get?
How far away from the spot it was
hit will it land?
At its highest point the final velocity of the ball in the y
direction will be 0. So we know initial and final velocity
and acceleration and time if we cut the three seconds in
half so there are several ways we could solve this;
V2f = V2i +2aΔy
Vf = 0
30°
26.0m/s
Vx
Vi =15m/s
a = -10m/s2
y = 11.3m
Breaking a Vector into Components.
A golf ball is hit eastward at an angle of 30° with
respect to the ground with an initial velocity of
30.0m/s. What are are the horizontal and vertical
components of the golf ball’s motion?
Now that you know the horizontal
and vertical components of motion
you could answer questions such
as:
•
•
•
15.0m/s
Vy
How long will the ball be in the air?
How high with ball get?
How far away from the spot it was
hit will it land?
Since we know the total time of flight and the velocity
of the ball in the horizontal direction we can use the
following formula:
Δx = vt
V = 26m/s t =3s
x = 78m
30°
26.0m/s
Vx
Determine the Components
For each image below determine the x and y components using trig.
[1]
40°
[3]
Y-component
Y-component
40°
X-component
X-component
[2]
[4]
Y-component
Y-component
30°
67°
X-component
X-component
Adding and Subtracting Vectors
•
•
•
•
It takes longer to fly to California than it does to fly back. Why?
Wind usually moves from West to East Across the United States (think
about where most of our storms come from!)
When you fly West you are usually flying into the wind.
When you fly East the wind is at your back pushing you.
Img via physicsclassroom
Img via physicsclassroom
Rules for adding and subtracting vectors
1)
Vectors must be parallel or in the same line to directly add or subtract them.
2)
If not you must break a vector into components first.
3)
Then you add and subtract vector components.
4)
Then you get the resultant of those components (tail to tip)
Think of vector addition in terms of tug of war or pushing and pulling a box.
10N
10N
10N
8N
Net force = 0
5N
5N
10N
Net force = 7N to the right
Net force = 5N to the right
10N
5N
Net force = 15N to the right
7N
3N
+
=
4N
Rules for adding and subtracting vectors
1)
Vectors must be parallel or in the same line to directly add or subtract them.
2)
If not you must break a vector into components first.
3)
Then you add and subtract vector components.
4)
Then you get the resultant of those components (tail to tip)
So now we have 3
component vectors
Two people are pulling a box as
depicted in the image on the left.
Find the Resultant
10N
45°
10N
10N
The red vector is in component form
and is fine. We need to determine
the components of the other vector
fist.
Add the red vectors:
17N
7N
Use tail to tip for resultant:
10N
= 7N
19N
Y component
45°
x component
7N
7N
7N
22°
= 7N
17N
Use trig to get the magnitude and angle
Tan-1 = 7N/17N = 22°
Sin(22°)= 7N/H
H=19N
Rules for adding and subtracting vectors
1)
Vectors must be parallel or in the same line to directly add or subtract them.
2)
If not you must break a vector into components first.
3)
Then you add and subtract vector components.
4)
Then you get the resultant of those components (tail to tip)
3.5N
3.5N
5N
Find the
Resultant
5N
Y component
Y component
5N
45°
x component
45°
5N
45°
x component
3.5N
45°
Our Four Components
Y = 3.5N north
X = 3.5N west
3.5N
Y = 3.5N north
X = 3.5N east
+
The end result is that the box moves as if one force
is pulling it to the North with a force of 7N.
=
7N
Resultant
Rules for adding and subtracting vectors
1)
Vectors must be parallel or in the same line to directly add or subtract them.
2)
If not you must break a vector into components first.
3)
Then you add and subtract vector components.
4)
Then you get the resultant of those components (tail to tip)
10N
[1]
Find the resultant
[3
]
9N
45°
10N
8N
Find the resultant
[2
]
40°
10N
10N
45°
45°
[4
]
Find the resultant
5N
7N
3N
8N
Find the resultant
40°
8N
40°
You have 5 asteroids all moving in the same direction
as a rocket ship with the speeds listed below.
A
B
C
D
E
NASA
600m/s
700m/s
800m/s
400m/s
600m/s
1. List all the asteroids that are moving towards the ship?
2. List all the asteroids that are moving away from the ship?
3. Explain your reasoning.
Modified from TIPERs
400m/s
A
B
Vp = 10m/s
Vp = 8m/s
Vt = 24m/s
Four cases where a
person is running on a
flatbed train. In cases C
and D the person is
running in the same
direction as the train. In
cases A and B the person
is running in a direction
opposite of the train’s
motion. An observer is
standing outside the
tracks watching.
Vt = 30m/s
C
D
Vp = 12m/s
Vp = 4m/s
Vt = 20m/s
Vt = 16m/s
Rank the velocity of the runners from the perspective of an outside observer standing besides the tracks. Explain.
or
1
Greatest
2
3
4
Least
All the
same
All Zero
Cannot
Determine
Modified from TIPERs
Ntipers Vector Question
2-SCT08: ADDING TWO VECTORS—MAGNITUDE OF THE RESULTANT
􏰀􏰀􏰀
Three students are discussing the magnitude of t he resultant of the addition of the vectors A and B.
Vector A has a 􏰀 magnitude of 5 cm, and vector B has a magnitude of 3 cm.
Alexis: “We’d have to know the directions of the vectors to know how big the resultant is going to be.”
Bert: “Since we are only asked about the magnitude, we don’ t have to worry about the directions. The
magnitude is just the size, so to find the magnitude of the resultant we just have to add the sizes of the
vectors. The magnitude of the resultant in this case is 8 cm.”
Cara: “No, these are vectors, and to find the magnitude you have to use the Pythagorean theorem. In
this case the magnitude is the square root of 34, a little less than 6 cm.”
Dacia: “The resultant is the vector that you have to add to the first vector to get the second vector. In
this case the resultant is 2.”
With which, if any, of these students do you agree?
Alexis _____ Bert _____ Cara _____ Dacia_____ None of them_____
Explain your reasoning.
(a) List all the velocities that have a zero x-component:
(b) List all the velocities that have a zero y-component:
(c) List all the velocities that have an x-component pointing in the positive x-direction:
(d) List all the velocities that have a y-component pointing in the negative y-direction:
Compressed String
If you simultaneously drop an object and shoot
an object horizontally from the same height,
which one hits the ground first?
0s
1s
2s
3s
They hit at the same time! The vertical component of
motion has nothing to do with how fast an object
falls. The only acceleration in that scenario is gravity
and it acts exclusively in the y direction.
4s
Vi
R = sin(2q )
g
2
Range
If you know initial velocity and the angle of
launch the range formula simplifies finding
horizontal displacement assuming the
following conditions are true:
1. A golf ball is driven off of a tee with
an initial velocity of 70m/s at an
angle of 12° with respect to the
ground. How far does it go?
2. A football is kicked with an initial
velocity of 23 m/s at an angle of 43°
with respect to the ground, how far
away will it land?
3. A long jumper has an initial velocity
of 11 m/s at an angle of 26°. How far
can he jump?
Assumptions
•
Earth is flat
•
Uniform gravitational field
•
No air resistance
•
Launch height = landing height.
• Range and Max Height
Img via https://share.ehs.uen.org
A ramp is elevated so that a ball rolls down it without slipping and then off of a table.
Just upon reaching the bottom of the ramp, the ball has a horizontal velocity of 5m/s.
You need to calculate where it will land and then place a cup in that spot. Then we will
test your calculations.
Angle of
the Ramp
Horizontal
Velocity
Average
Vx
20°
30°
40°
50°
Horizontal Displacement:
Δx = Vixt
Vertical Free Fall Time:
Img via https://share.ehs.uen.org
Δy = Viyt + ½at2
Calculate the horizontal velocity of a projectile that rolls down an inclined plane with our
different elevations. Conduct 5 trials for each angle and get an average value. Make sure
you redo any “outliers”.
Create a graph of elevation angle vs. horizontal velocity. Is it linear? Can you infer
anything about the angle of an inclined plane and acceleration?
Img via https://share.ehs.uen.org
A ball rolls down a 1m down a ramp elevated at 45°
without slipping and then falls off a 1m high table. Where
it will land?
Bullet Drop in Battlefield vs CoD
• None in CoD
• Battlefield is not necessary more accurate.
• Sure they have bullet drop but they often inflate
it due to small map sizes.
• Game files show g as being 15m/s2
• Why? When playing TDM type games the maps
are so small and the guns so powerful and have
such a high muzzle velocity that the bullet drop
has to be exaggerated to be relevant.