Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking...
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![Page 1: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/1.jpg)
Chapter 12 Kinematics
![Page 2: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/2.jpg)
ME 242 Chapter 12
Question 1 We obtain the acceleration fastest
(A)By taking the derivative of x(t)(B)By Integrating x(t) twice(C)By integrating the accel as function of displacement(D)By computing the time to liftoff, then choosing the
accel such that the velocity is 160 mph
![Page 3: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/3.jpg)
![Page 4: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/4.jpg)
Question 2 The acceleration is approximately
(A)92 ft/s2(B)66 ft/s2
(C)85.3 ft/s2(D)182 ft/s2(E)18.75 ft/s2
ME 242 Chapter 12
Ya pili
160 mi/h = 235 ft/s
![Page 5: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/5.jpg)
Question 2 The acceleration is approximately
(A)92 ft/s2
(B)66 ft/s2(C)85.3 ft/s2(D)182 ft/s2(E) 18.75 ft/s2
ME 242 Chapter 12
Ya kwanza
Solution:160 mi/h = 235 ft/sWe use v*dv = a*dxIntegrate: 1/2v2 = a*d,
where d is the length of the runway, and the start velocity = 0
22
22
/92300
235*5.0
*300*5.0
sfta
aftv
![Page 6: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/6.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 7: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/7.jpg)
![Page 8: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/8.jpg)
ME242 Tutoring
• Graduate Assistant Ms. Yang Liu will be available to assist with homework preparation and answer questions.
• Coordination through the Academic Success Center in TBE-A 207 Tuesday and Friday mornings.
• Contact hours: MW after class
![Page 9: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/9.jpg)
ME242 Reading Assignments
• Look up the next Homework assignment on Mastering
• Example: your second assignment covers sections 12.5 and 12.6
• Study the text and practice the examples in the book
• An I-Clicker reading test on each chapter section will be given at the start of each lecture
• More time for discussion and examples
![Page 10: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/10.jpg)
Supplemental Instruction ME 242
• Questions – Yang Liu – PhD student in ME– [email protected]– Lab: SEB 4261
![Page 11: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/11.jpg)
A (x0,y0)
B (d,h)v
0g
horiz.
distance = dx
yh
Chapter 12-5 Curvilinear Motion X-Y Coordinates
![Page 12: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/12.jpg)
![Page 13: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/13.jpg)
Here is the solution in Mathcad
![Page 14: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/14.jpg)
Example: Hit target at Position (360’, -80’)
![Page 15: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/15.jpg)
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)
![Page 16: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/16.jpg)
NORMAL AND TANGENTIAL COMPONENTS (Section 12.7)
When a particle moves along a curved path, it is sometimes convenient to describe its motion using coordinates other than Cartesian. When the path of motion is known, normal (n) and tangential (t) coordinates are often used.
In the n-t coordinate system, the origin is located on the particle (the origin moves with the particle).
The t-axis is tangent to the path (curve) at the instant considered, positive in the direction of the particle’s motion.The n-axis is perpendicular to the t-axis with the positive direction toward the center of curvature of the curve.
![Page 17: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/17.jpg)
NORMAL AND TANGENTIAL COMPONENTS (continued)
The position of the particle at any instant is defined by the distance, s, along the curve from a fixed reference point.
The positive n and t directions are defined by the unit vectors un and ut, respectively.
The center of curvature, O’, always lies on the concave side of the curve.The radius of curvature, r, is defined as the perpendicular distance from the curve to the center of curvature at that point.
![Page 18: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/18.jpg)
ACCELERATION IN THE n-t COORDINATE SYSTEM
Acceleration is the time rate of change of velocity:a = dv/dt = d(vut)/dt = vut + vut
. .
Here v represents the change in the magnitude of velocity and ut represents the rate of change in the direction of ut.
..
. a = v ut + (v2/r) un = at ut + an un.
After mathematical manipulation, the acceleration vector can be expressed as:
![Page 19: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/19.jpg)
ACCELERATION IN THE n-t COORDINATE SYSTEM (continued)
So, there are two components to the acceleration vector:
a = at ut + an un
• The normal or centripetal component is always directed toward the center of curvature of the curve. an = v2/r
• The tangential component is tangent to the curve and in the direction of increasing or decreasing velocity.
at = v or at ds = v dv.
• The magnitude of the acceleration vector is a = [(at)2 + (an)2]0.5
![Page 20: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/20.jpg)
NORMAL AND TANGENTIAL COMPONENTS (Section 12.7)
When a particle moves along a curved path, it is sometimes convenient to describe its motion using coordinates other than Cartesian. When the path of motion is known, normal (n) and tangential (t) coordinates are often used.
In the n-t coordinate system, the origin is located on the particle (the origin moves with the particle).
The t-axis is tangent to the path (curve) at the instant considered, positive in the direction of the particle’s motion.The n-axis is perpendicular to the t-axis with the positive direction toward the center of curvature of the curve.
![Page 21: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/21.jpg)
Normal and Tangential CoordinatesVelocity Page 53 tusv *
![Page 22: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/22.jpg)
Normal and Tangential Coordinates
‘e’ denotes unit vector(‘u’ in Hibbeler)
![Page 23: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/23.jpg)
‘e’ denotes unit vector(‘u’ in Hibbeler)
![Page 24: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/24.jpg)
Learning Techniques
•Complete Every Homework
•Team with fellow students
•Study the Examples
•Ask: Ms. Yang, peers, me•Mathcad provides structure and numerically correct results
![Page 25: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/25.jpg)
Course Concepts•Math
•Think Conceptually
•Map your approach BEFORE starting work
![Page 26: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/26.jpg)
12.8 Polar coordinates
![Page 27: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/27.jpg)
Polar coordinates
‘e’ denotes unit vector(‘u’ in Hibbeler)
![Page 28: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/28.jpg)
Polar coordinates
‘e’ denotes unit vector(‘u’ in Hibbeler)
![Page 29: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/29.jpg)
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![Page 30: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/30.jpg)
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![Page 31: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/31.jpg)
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![Page 32: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/32.jpg)
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.)
![Page 33: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/33.jpg)
Vectors and Geometry
j
ix
y
(q t)
r(t)
q
![Page 34: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/34.jpg)
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 35: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/35.jpg)
Example Vector equation: Sailboat tacking at 50 deg. against Northern Wind
(blue vector)
BoatWindBoatWind VVV /
We solve Graphically (Vector Addition)
![Page 36: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/36.jpg)
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
![Page 37: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/37.jpg)
ABAB VVV /
Plane Vector Addition is two-dimensional.
12.10 Relative (Constrained) Motion
vB
vA
vB/A
![Page 38: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/38.jpg)
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
![Page 39: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/39.jpg)
Chapter 12.10 Relative Motion
![Page 40: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/40.jpg)
BABA rrr /
Vector Addition
![Page 41: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/41.jpg)
BABA VVV /
Differentiating gives:
![Page 42: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/42.jpg)
ABAB VVV /
![Page 43: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/43.jpg)
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
![Page 44: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/44.jpg)
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.
![Page 45: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/45.jpg)
Exam 1
Preparation: Practice: Step 1: Describe Problem Mathematically
Step2: Calculus and Algebraic Equation Solving
![Page 46: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/46.jpg)
And here a few visual observations about contemporary forms of socializing, sent to me by a colleague at the Air Force Academy.Enjoy!
![Page 47: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/47.jpg)
Having coffee with friends.
![Page 48: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/48.jpg)
A day at the beach.
![Page 49: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/49.jpg)
Cheering on your team.
![Page 50: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/50.jpg)
Out on an intimate date.
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Having a conversation with your BFF
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A visit to the museum
![Page 53: Chapter 12 Kinematics. ME 242 Chapter 12 Question 1 We obtain the acceleration fastest (A)By taking the derivative of x(t) (B)By Integrating x(t) twice.](https://reader033.fdocuments.net/reader033/viewer/2022051619/56649e4d5503460f94b440f3/html5/thumbnails/53.jpg)
Enjoying the sights