Chapter 3 Kinematics in Two Dimensions; Vectors 1.
-
Upload
raymond-hoover -
Category
Documents
-
view
240 -
download
0
description
Transcript of Chapter 3 Kinematics in Two Dimensions; Vectors 1.
![Page 1: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/1.jpg)
Chapter 3
Kinematics in Two Dimensions; Vectors
1
![Page 2: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/2.jpg)
3-1 Vectors and Scalars
A vector has magnitude as well as direction. (displacement, velocity, force, momentum)
A scalar has only a magnitude. (mass, time, temperature)
2
![Page 3: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/3.jpg)
3-2 Addition of Vectors – Graphical Methods
For vectors in one dimension, simple addition and subtraction are all that is needed.
You do need to be careful about the signs, as the figure indicates.
3
![Page 4: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/4.jpg)
Addition of vectors that are vertical or horizontal only
4
![Page 5: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/5.jpg)
If the motion is in two dimensions, the situation gets more complicated.
5
![Page 6: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/6.jpg)
2 Dimensional Kinematics• We need to use
vector diagrams to describe the motion
6
![Page 7: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/7.jpg)
Describing the direction or angle
of the vector7
![Page 8: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/8.jpg)
Describing the magnitude of the vector
8
![Page 9: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/9.jpg)
3-2 Addition of Vectors – Graphical Methods
Even if the vectors are not at right angles, they can be added graphically using the “tail-to-tip” method.
9
![Page 10: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/10.jpg)
10
![Page 11: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/11.jpg)
3-2 Addition of Vectors – Graphical MethodsIf the paths are at right angles to one another; we can find the resultant by using the Pythagorean Theorem.
11
![Page 12: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/12.jpg)
Using the Pythagorean Theorem to find the magnitude of the vector
12
![Page 13: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/13.jpg)
3-4 Adding Vectors by Components
If the components are perpendicular, they can be found using trigonometric functions.
Remember: SOH-CAH-TOA! 13
![Page 14: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/14.jpg)
To determine the direction or angle of the resultant vector
14
Tan 111
Tan 1)
![Page 15: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/15.jpg)
15
![Page 16: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/16.jpg)
Vertical vector components are noted with a y and horizontal with an x
16
![Page 17: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/17.jpg)
17
Fx
Fy
Fx
Fx
Fx
Fy
Fy
Fy
![Page 18: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/18.jpg)
3-4 Adding Vectors by Components
Adding vectors:1. Draw a diagram.
2. Choose x and y axes.
3. Resolve each vector into x and y components.
4. Calculate each component using trig functions.
5. Add the components in each direction.
6. To find the magnitude and direction of the vector, use:
18
![Page 19: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/19.jpg)
3-5 Projectile MotionA projectile is an object moving in two dimensions under the influence of Earth's gravity; its path is a parabola.
19
![Page 20: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/20.jpg)
20
![Page 21: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/21.jpg)
Can be understood by analyzing the horizontal and vertical motions separately. The x and y components are independent of each other!
21
![Page 22: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/22.jpg)
HorizontalMotion (x)
VerticalMotion (y)
Forces(Present? - Yes or No. If present, what direction?)
NoYes The force of gravity acts downward
Acceleration(Present? - Yes or No. If present, what direction?)
No Yes "g" is downward at 9.8 m/s/s
Velocity(Constant or Changing?)
Constant Changing (by 9.8 m/s each second)
Analyzing parabolic motion
22
![Page 23: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/23.jpg)
23
![Page 24: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/24.jpg)
At 1 sec time intervals
24
![Page 25: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/25.jpg)
3-5 Projectile MotionIf an object is launched at an initial angle of θ0 with the horizontal, the analysis is similar except that the initial velocity has a vertical component.
25
![Page 26: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/26.jpg)
Objects launched at an angle
26
![Page 27: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/27.jpg)
Our previously learned equations still work! We just have to analyze the x and y components of the
motion separately.
27
![Page 28: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/28.jpg)
3-6 Solving Problems Involving Projectile Motion
1. Read the problem carefully, and choose the object(s) you are going to analyze.
2. Draw a diagram.
3. Choose an origin and a coordinate system.
4. Decide on the time interval; this is the same in both directions, and includes only the time the object is moving with constant acceleration g.
5. Examine the x and y motions separately.
6. List known and unknown quantities. Remember that vx never changes, and that vy = 0 at the highest point.
7. Plan how you will proceed. Use the appropriate equations; you may have to combine some of them.
28
![Page 29: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/29.jpg)
It’s all relative!
29
![Page 30: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/30.jpg)
Magnitude of the resultantA2 + B2 = R2
(100 km/hr)2 + (25 km/hr)2 = R2 R= 103.1 km/hr
Direction of resultanttan Ө= opp/adjtan Ө = (25/100)Ө = 14.0 degrees
30
![Page 31: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/31.jpg)
Each velocity is labeled first with the object, and second with the reference frame in which it has this velocity. Therefore, vWS is the velocity of the water in the shore frame, vBS is the velocity of the boat in the shore frame, and vBW is the velocity of the boat in the water frame.
31
![Page 32: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/32.jpg)
3-8 Relative VelocityIn this case, the relationship between the three velocities is:
(3-6)
32
![Page 33: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/33.jpg)
33
![Page 34: Chapter 3 Kinematics in Two Dimensions; Vectors 1.](https://reader035.fdocuments.net/reader035/viewer/2022081502/5a4d1b147f8b9ab059990c1c/html5/thumbnails/34.jpg)
References
Giancoli, Douglas. Physics: Principles with Applications 6th
Edition. 2009. http://www.physicsclassroom.com
34