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

...

...

...

...

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