Question 3 Road map: We obtain the velocity fastest (A)By Taking the derivative of a(t) (B)By...
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![Page 1: Question 3 Road map: We obtain the velocity fastest (A)By Taking the derivative of a(t) (B)By Integrating a(t) (C)By integrating the accel as function.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d565503460f94a337f2/html5/thumbnails/1.jpg)
Question 3 Road map: We obtain the velocity fastest
(A)By Taking the derivative of a(t)(B)By Integrating a(t)(C)By integrating the accel as function of displacement(D)By computing the time to bottom, then computing the
velocity.
![Page 2: Question 3 Road map: We obtain the velocity fastest (A)By Taking the derivative of a(t) (B)By Integrating a(t) (C)By integrating the accel as function.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d565503460f94a337f2/html5/thumbnails/2.jpg)
![Page 3: Question 3 Road map: We obtain the velocity fastest (A)By Taking the derivative of a(t) (B)By Integrating a(t) (C)By integrating the accel as function.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d565503460f94a337f2/html5/thumbnails/3.jpg)
Question 3 Road map: We obtain the velocity fastest
(A)By Taking the derivative of a(t)(B)By Integrating a(t)(C)By integrating the accel as function of displacement(D)By computing the time to bottom, then computing the
velocity.
![Page 4: Question 3 Road map: We obtain the velocity fastest (A)By Taking the derivative of a(t) (B)By Integrating a(t) (C)By integrating the accel as function.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d565503460f94a337f2/html5/thumbnails/4.jpg)
A (x0,y0)
B (d,h)v
0g
horiz.
distance = dx
yh
Chapter 12-5 Curvilinear Motion X-Y Coordinates
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![Page 6: Question 3 Road map: We obtain the velocity fastest (A)By Taking the derivative of a(t) (B)By Integrating a(t) (C)By integrating the accel as function.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d565503460f94a337f2/html5/thumbnails/6.jpg)
Here is the solution in Mathcad
![Page 7: Question 3 Road map: We obtain the velocity fastest (A)By Taking the derivative of a(t) (B)By Integrating a(t) (C)By integrating the accel as function.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d565503460f94a337f2/html5/thumbnails/7.jpg)
Example: Hit target at Position (360’, -80’)
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0 100 200 300100
50
0
50
92.87
100
h1 t( )
h2 t( )
3600 d1 t( ) d2 t( )
Two solutions exist (Tall Trajectory and flat Trajectory).The Given - Find routine finds only one solution, depending on the guessvalues chosen. Therefore we must solve twice, using multiple guessvalues. We can also solve explicitly, by inserting one equation into thesecond:
Example: Hit target at Position (360, -80)
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12.7 Normal and Tangential Coordinatesut : unit tangent to the pathun : unit normal to the path
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Normal and Tangential CoordinatesVelocity Page 53 tusv *
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Normal and Tangential Coordinates
‘e’ denotes unit vector(‘u’ in Hibbeler)
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‘e’ denotes unit vector(‘u’ in Hibbeler)
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12.8 Polar coordinates
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Polar coordinates
‘e’ denotes unit vector(‘u’ in Hibbeler)
![Page 15: Question 3 Road map: We obtain the velocity fastest (A)By Taking the derivative of a(t) (B)By Integrating a(t) (C)By integrating the accel as function.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d565503460f94a337f2/html5/thumbnails/15.jpg)
Polar coordinates
‘e’ denotes unit vector(‘u’ in Hibbeler)
![Page 16: Question 3 Road map: We obtain the velocity fastest (A)By Taking the derivative of a(t) (B)By Integrating a(t) (C)By integrating the accel as function.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d565503460f94a337f2/html5/thumbnails/16.jpg)
12.8 Polar coordinates
In a polar coordinate system, the velocity vector can be written as v = vrur + vθuθ = rur +ru. The term is called
A) transverse velocity.
B) radial velocity.
C) angular velocity.
D) angular acceleration
...
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12.10 Relative (Constrained) Motion
L
B
A
i
J
vA = const
vA is given as shown.Find vB
Approach: Use rel. Velocity:vB = vA +vB/A
(transl. + rot.)
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Vectors and Geometry
j
ix
y
t
r(t)
![Page 22: Question 3 Road map: We obtain the velocity fastest (A)By Taking the derivative of a(t) (B)By Integrating a(t) (C)By integrating the accel as function.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d565503460f94a337f2/html5/thumbnails/22.jpg)
A
ResultB
Given: vectors A and B as shown. The RESULT vector is:•(A) RESULT = A - B
•(B) RESULT = A + B•(C) None of the above
![Page 23: Question 3 Road map: We obtain the velocity fastest (A)By Taking the derivative of a(t) (B)By Integrating a(t) (C)By integrating the accel as function.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d565503460f94a337f2/html5/thumbnails/23.jpg)
A
ResultB
Given: vectors A and B as shown. The RESULT vector is:•(A) RESULT = A - B
•(B) RESULT = A + B•(C) None of the above
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Make a sketch: A V_rel
v_Truck
BThe rel. velocity is:
V_Car/Truck = v_Car -vTruck
12.10 Relative (Constrained) Motion
V_truck = 60V_car = 65
![Page 25: Question 3 Road map: We obtain the velocity fastest (A)By Taking the derivative of a(t) (B)By Integrating a(t) (C)By integrating the accel as function.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d565503460f94a337f2/html5/thumbnails/25.jpg)
Make a sketch: A V_river
v_boat
B The velocity is:(A)V_total = v+boat – v_river(B)V_total = v+boat + v_river
12.10 Relative (Constrained) Motion
![Page 26: Question 3 Road map: We obtain the velocity fastest (A)By Taking the derivative of a(t) (B)By Integrating a(t) (C)By integrating the accel as function.](https://reader030.fdocuments.net/reader030/viewer/2022032704/56649d565503460f94a337f2/html5/thumbnails/26.jpg)
Make a sketch: A V_river
v_boat
B The velocity is:(A)V_total = v+boat – v_river(B)V_total = v+boat + v_river
12.10 Relative (Constrained) Motion
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Rel. Velocity example: Solution
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Example Vector equation: Sailboat tacking at 50 deg. against Northern Wind
(blue vector)
BoatWindBoatWind VVV /
We solve Graphically (Vector Addition)
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Example Vector equation: Sailboat tacking at 50 deg. against Northern Wind
BoatWindBoatWind VVV /
An observer on land (fixed Cartesian Reference) sees Vwind and vBoat .
Land
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ABAB VVV /
Plane Vector Addition is two-dimensional.
12.10 Relative (Constrained) Motion
vB
vA
vB/A
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Example cont’d: Sailboat tacking against Northern Wind
BoatWindBoatWind VVV /
2. Vector equation (1 scalar eqn. each in i- and j-direction). Solve using the given data (Vector Lengths and orientations) and Trigonometry
500
150
i
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Chapter 12.10 Relative Motion
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BABA rrr /
Vector Addition
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BABA VVV /
Differentiating gives:
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ABAB VVV /
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Exam 1• We will focus on Conceptual Solutions. Numbers are secondary.• Train the General Method• Topics: All covered sections of Chapter 12• Practice: Train yourself to solve all Problems in Chapter 12
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Exam 1
Preparation: Start now! Cramming won’t work.
Questions: Discuss with your peers. Ask me.
The exam will MEASURE your knowledge and give you objective feedback.
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Exam 1
Preparation: Practice: Step 1: Describe Problem Mathematically
Step2: Calculus and Algebraic Equation Solving