Post on 21-Dec-2015
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.
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.
A (x0,y0)
B (d,h)v
0g
horiz.
distance = dx
yh
Chapter 12-5 Curvilinear Motion X-Y Coordinates
Here is the solution in Mathcad
Example: Hit target at Position (360’, -80’)
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)
12.7 Normal and Tangential Coordinatesut : unit tangent to the pathun : unit normal to the path
Normal and Tangential CoordinatesVelocity Page 53 tusv *
Normal and Tangential Coordinates
‘e’ denotes unit vector(‘u’ in Hibbeler)
‘e’ denotes unit vector(‘u’ in Hibbeler)
12.8 Polar coordinates
Polar coordinates
‘e’ denotes unit vector(‘u’ in Hibbeler)
Polar coordinates
‘e’ denotes unit vector(‘u’ in Hibbeler)
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.)
Vectors and Geometry
j
ix
y
t
r(t)
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
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
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
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
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
Rel. Velocity example: Solution
Example Vector equation: Sailboat tacking at 50 deg. against Northern Wind
(blue vector)
BoatWindBoatWind VVV /
We solve Graphically (Vector Addition)
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
ABAB VVV /
Plane Vector Addition is two-dimensional.
12.10 Relative (Constrained) Motion
vB
vA
vB/A
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
Chapter 12.10 Relative Motion
BABA rrr /
Vector Addition
BABA VVV /
Differentiating gives:
ABAB VVV /
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
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.
Exam 1
Preparation: Practice: Step 1: Describe Problem Mathematically
Step2: Calculus and Algebraic Equation Solving