Chapter 3 Motion in two or more dimensions. Two dimensional motion.
Motion in Two and Three Dimensions - University of Louisville
Transcript of Motion in Two and Three Dimensions - University of Louisville
1Prof. Sergio B. MendesSummer 2018
Chapter 3 of Essential University Physics, Richard Wolfson, 3rd Edition
Motion in Two and Three Dimensions
2Prof. Sergio B. MendesSummer 2018
Physical Quantities
VectorsScalars
β’ Temperature β’ Displacement
β’ Mass
β’ Pressure
β’ Velocity
β’ Acceleration
β’ Forceβ’ Volume
Magnitude (number & unit) Magnitude (number & unit) and Direction
3Prof. Sergio B. MendesSummer 2018
Summing Two Vectors
π¨π¨
π©π©
π¨π¨ + π©π© = πͺπͺ
π©π©
π¨π¨πͺπͺ
π¨π¨ + π©π© =
πΆπΆ = π΄π΄2 + π΅π΅2 + 2 π΄π΄ π΅π΅ ππππππ πΌπΌ
πΌπΌ
π¨π¨π΄π΄ = ππππππππππππππππππ ππππ
π©π©π΅π΅ = ππππππππππππππππππ ππππ
πͺπͺπΆπΆ = ππππππππππππππππππ ππππ
π©π©
π¨π¨
π©π© + π¨π¨ = πͺπͺ
4Prof. Sergio B. MendesSummer 2018
Summing Three Vectors
π¨π¨
π©π©
π¨π¨ + π©π© + πͺπͺ = π«π«
πͺπͺ
π«π«
π¨π¨
π©π©
πͺπͺ
π¨π¨ + π©π© + πͺπͺ
π¨π¨ + π©π©π©π© + πͺπͺ
π«π«
= π«π«
5Prof. Sergio B. MendesSummer 2018
Subtracting Two Vectors
π¨π¨
π¨π¨ β π©π© = πͺπͺ
πͺπͺ
πΆπΆ = π΄π΄2 + π΅π΅2 β 2 π΄π΄ π΅π΅ ππππππ πΌπΌ
πΌπΌ
π©π©
π¨π¨
βπ©π©
βπ©π©
6Prof. Sergio B. MendesSummer 2018
Multiplying a Vector by a Scalarπ¨π¨ π½π½
π½π½ π¨π¨
π¨π¨
π¨π¨
π½π½ π¨π¨
If π½π½ is positive
If π½π½ is negative
7Prof. Sergio B. MendesSummer 2018
Representing a Vector in a Cartesian Coordinate System: 2D
π΄π΄π₯π₯ = π΄π΄ ππππππ ππ
π΄π΄π¦π¦ = π΄π΄ ππππππ ππ
8Prof. Sergio B. MendesSummer 2018
Representing a Vector in terms of Unit Vectors: 2D
π΄π΄π₯π₯ = π΄π΄ ππππππ ππ
π΄π΄π¦π¦ = π΄π΄ ππππππ ππ
π¨π¨ = π΄π΄π₯π₯ οΏ½ΜοΏ½π + π΄π΄π¦π¦ οΏ½ΜοΏ½π
9Prof. Sergio B. MendesSummer 2018
Representation of a Vector in terms of Unit Vectors: 3D
π¨π¨ = π΄π΄π₯π₯ οΏ½ΜοΏ½π + π΄π΄π¦π¦ οΏ½ΜοΏ½π + π΄π΄π§π§ οΏ½ππ
10Prof. Sergio B. MendesSummer 2018
Example 3.1
11Prof. Sergio B. MendesSummer 2018
Position Vector
ππ β‘ π₯π₯ οΏ½ΜοΏ½π + π¦π¦ οΏ½ΜοΏ½π + π§π§ οΏ½ππππ
π₯π₯
π¦π¦
π§π§
πͺπͺ
12Prof. Sergio B. MendesSummer 2018
Displacement Vectorβππ β‘ ππππ β ππππ
π₯π₯
π¦π¦
π§π§
πͺπͺ
ππππππππ
βπ₯π₯ = π₯π₯2 β π₯π₯1
βπ¦π¦ = π¦π¦2 β π¦π¦1
βπ§π§ = π§π§2 β π§π§1
βππ = ππππ β ππππ
βπ₯π₯ οΏ½ΜοΏ½π + βπ¦π¦ οΏ½ΜοΏ½π + βπ§π§ οΏ½ππ = π₯π₯2 β π₯π₯1 οΏ½ΜοΏ½π + π¦π¦2 β π¦π¦1 οΏ½ΜοΏ½π + π§π§2 β π§π§1 οΏ½ππ
= ππππ + βππ
13Prof. Sergio B. MendesSummer 2018
Average Velocity VectorοΏ½ππ β‘
ππππ β ππππππ2 β ππ1
π₯π₯
π¦π¦
π§π§
πͺπͺ
ππππ
οΏ½Μ οΏ½π£π₯π₯ =π₯π₯2 β π₯π₯1ππ2 β ππ1
=βπ₯π₯βππ
οΏ½Μ οΏ½π£π¦π¦ =π¦π¦2 β π¦π¦1ππ2 β ππ1
=βπ¦π¦βππ
οΏ½Μ οΏ½π£π§π§ =π§π§2 β π§π§1ππ2 β ππ1
=βπ§π§βππ
βππ = ππππ β ππππ = οΏ½ππ ππ2 β ππ1
ππππ = ππππ + βππ
οΏ½Μ οΏ½π£π₯π₯ οΏ½ΜοΏ½π + οΏ½Μ οΏ½π£π¦π¦ οΏ½ΜοΏ½π + οΏ½Μ οΏ½π£π§π§ οΏ½ππ = =βππβππ
14Prof. Sergio B. MendesSummer 2018
Instantaneous Velocity Vector
ππ β‘ limβπ‘π‘βππ
βππβππ
π£π£π₯π₯ = limβπ‘π‘βππ
βπ₯π₯βππ
=πππ₯π₯ππππ
π£π£π₯π₯ οΏ½ΜοΏ½π + π£π£π¦π¦ οΏ½ΜοΏ½π + π£π£π§π§ οΏ½ππ =
π£π£π¦π¦ = limβπ‘π‘βππ
βπ¦π¦βππ
=πππ¦π¦ππππ
π£π£π§π§ = limβπ‘π‘βππ
βπ§π§βππ
=πππ§π§ππππ
15Prof. Sergio B. MendesSummer 2018
Average Acceleration VectorοΏ½ππ β‘
ππππ β ππππππ2 β ππ1
π₯π₯
π¦π¦
π§π§
πͺπͺ
ππππ
οΏ½πππ₯π₯ =π£π£2 β π£π£1ππ2 β ππ1
=βπ£π£βππ
οΏ½πππ¦π¦ =π£π£2 β π£π£1ππ2 β ππ1
=βπ£π£βππ
οΏ½πππ§π§ =π£π£2 β π£π£1ππ2 β ππ1
=βπ§π§βππ
βππ = οΏ½ππ ππ2 β ππ1
ππππ = ππππ + βππ
οΏ½πππ₯π₯ οΏ½ΜοΏ½π + οΏ½πππ¦π¦ οΏ½ΜοΏ½π + οΏ½πππ§π§ οΏ½ππ = =βππβππ
16Prof. Sergio B. MendesSummer 2018
Instantaneous Acceleration Vector
ππ β‘ limβπ‘π‘βππ
βππβππ
πππ₯π₯ = limβπ‘π‘βππ
βπ£π£π₯π₯βππ
=πππ£π£π₯π₯ππππ
πππ₯π₯ οΏ½ΜοΏ½π + πππ¦π¦ οΏ½ΜοΏ½π + πππ§π§ οΏ½ππ =
πππ¦π¦ = limβπ‘π‘βππ
βπ£π£π¦π¦βππ
=πππ£π£π¦π¦ππππ
πππ§π§ = limβπ‘π‘βππ
βπ£π£π§π§βππ
=πππ£π£π§π§ππππ
17Prof. Sergio B. MendesSummer 2018
A Few Observations
18Prof. Sergio B. MendesSummer 2018
Same Direction ππππ & ππ
Magnitude of velocity changes but not the direction.
19Prof. Sergio B. MendesSummer 2018
Opposite Direction ππππ & ππ
Magnitude of velocity changes and eventually the direction may be reversed.
20Prof. Sergio B. MendesSummer 2018
Arbitrary Direction ππππ & ππ
Direction and magnitude of velocity change.
21Prof. Sergio B. MendesSummer 2018
Velocity is always tangential to the trajectory
ππ β‘ limβπ‘π‘βππ
βππβππ ππ β₯ βππ
A curved (non-straight) trajectory always requires acceleration !!
22Prof. Sergio B. MendesSummer 2018
Relative Motion
βπππππππ‘π‘, ππππππππ
βππππππππππ, ππππππππ
βπππππππ‘π‘, ππππππππ = βπππππππ‘π‘, ππππππππ + βππππππππππ, ππππππππ
βπππππππ‘π‘, ππππππππ
βππ=βπππππππ‘π‘, ππππππππ
βππ+βππππππππππ, ππππππππ
βππlimβπ‘π‘βππ
πππππππ‘π‘, ππππππππ = πππππππ‘π‘, ππππππππ + ππππππππππ, ππππππππ
23Prof. Sergio B. MendesSummer 2018
Example 3.2A jetliner flies at 960 km/h relative to the air in a wind blowing eastward at 190 km/h. It wants to go 1290 km straight northward. In what direction should the plane point to track northward ? How long will the trip take ?
ππππ, ππ = ππππ, ππ + ππππ, ππ
ππππ, ππππππ, ππ
ππππ, ππ
ππππ, ππ = π£π£ππ, ππ οΏ½ΜοΏ½π = 190ππππβ
οΏ½ΜοΏ½π
οΏ½ΜοΏ½π
οΏ½ΜοΏ½π
π£π£ππ, ππ = 960ππππβ
ππππ, ππ = 0 οΏ½ΜοΏ½π + π£π£ππ, ππ οΏ½ΜοΏ½π
ππππ, ππ = π£π£ππ, ππ ππππππ ππ οΏ½ΜοΏ½π + π£π£ππ, ππ ππππππ ππ οΏ½ΜοΏ½π
What do we know ?
What do we want ?ππ = cosβ1
βπ£π£ππ, ππ
π£π£ππ, ππ= 101.4Β°
βππ =βπ¦π¦π£π£ππ, ππ
=βπ¦π¦
π£π£ππ, ππ ππππππ ππ= 1.4 h
βπ¦π¦ = 1290 ππππ
24Prof. Sergio B. MendesSummer 2018
Constant Acceleration in 3D
ππ ππ = ππ
πππ₯π₯ ππ = πππ₯π₯
πππ¦π¦ ππ = πππ¦π¦
πππ§π§ ππ = πππ§π§
25Prof. Sergio B. MendesSummer 2018
From the definition of average velocity in 3D:
βππ = ππ ππ2 β ππ ππ1 = οΏ½ππ Γ ππ2 β ππ1
ππ ππ β ππππ = ππ Γ ππ β 0
οΏ½ππ = ππ
ππ2 β‘ ππ
ππ1 β‘ 0
ππ ππ = ππππ + ππ ππ
ππ ππ1 β‘ 0 β‘ ππππ
26Prof. Sergio B. MendesSummer 2018
ππ ππ = ππππ + ππ ππ
π£π£π₯π₯ ππ = π£π£ππ,π₯π₯ + πππ₯π₯ ππ
π£π£π¦π¦ ππ = π£π£ππ,π¦π¦ + πππ¦π¦ ππ
π£π£π§π§ ππ = π£π£ππ,π§π§ + πππ§π§ ππ
27Prof. Sergio B. MendesSummer 2018
π£π£π₯π₯ = π£π£ππ,π₯π₯ + πππ₯π₯ ππ π₯π₯ ππ = π₯π₯ππ + π£π£ππ,π₯π₯ ππ +12πππ₯π₯ ππ2
π£π£π¦π¦ = π£π£ππ,π¦π¦ + πππ¦π¦ ππ π¦π¦ ππ = π¦π¦ππ + π£π£ππ,π¦π¦ ππ +12πππ¦π¦ ππ2
π£π£π§π§ = π£π£ππ,π§π§ + πππ§π§ ππ π§π§ ππ = π§π§ππ + π£π£ππ,π§π§ ππ +12πππ§π§ ππ2
ππ ππ = ππππ + ππππππ +12ππ ππππ
ππ ππ = π₯π₯ ππ οΏ½ΜοΏ½π + π¦π¦ ππ οΏ½ΜοΏ½π + π§π§ ππ οΏ½ππ
ππππ = π£π£ππ,π₯π₯ οΏ½ΜοΏ½π + π£π£ππ,π¦π¦ οΏ½ΜοΏ½π + π£π£ππ,π§π§ οΏ½ππ
ππ = πππ₯π₯ οΏ½ΜοΏ½π + πππ₯π₯ οΏ½ΜοΏ½π + πππ₯π₯ οΏ½ππ
ππππ = π₯π₯ππ οΏ½ΜοΏ½π + π¦π¦ππ οΏ½ΜοΏ½π + π§π§ππ οΏ½ππ
οΏ½ΜοΏ½π
οΏ½ΜοΏ½π
οΏ½ππ
28Prof. Sergio B. MendesSummer 2018
ππ ππ = ππππ + ππ ππ
ππ ππ = ππ
ππ ππ = ππππ + ππππ ππ +12ππ ππππ
Constant Acceleration in 3D, in Summary:
29Prof. Sergio B. MendesSummer 2018
Bottom Line: We can study the motion in each Cartesian direction independently.
30Prof. Sergio B. MendesSummer 2018
Example 3.3
π£π£ππ,π₯π₯ = 7.3 ππ/ππ πππ₯π₯ = ππ ππππππ 60Β°
πππ¦π¦ = ππ ππππππ 60Β°
ππ = 0.82 ππ/ππ2π£π£ππ = 7.3 ππ/ππ
π£π£ππ,π¦π¦ = 0
Youβre windsurfing at 7.3 m/s when a gust hits, accelerating your sailboard at 0.82 m/s2 at 60Β° to your original direction. If the gust last 8.7 s, whatβs the boardβs displacement during this time?
Ξππ = 8.7 ππ
31Prof. Sergio B. MendesSummer 2018
π₯π₯ ππ β π₯π₯ππ = π£π£ππ,π₯π₯ ππ +12πππ₯π₯ ππ2
π£π£ππ,π₯π₯ = 7.3 ππ/ππ
πππ₯π₯ = ππ ππππππ 60Β°
ππ = 0.82 ππ/ππ2
ππ = 8.7 ππ
π₯π₯ ππ = 8.7 ππ β π₯π₯ππ = 79.0 ππ
32Prof. Sergio B. MendesSummer 2018
π¦π¦ ππ β π¦π¦ππ = π£π£ππ,π¦π¦ ππ +12πππ¦π¦ ππ2
π£π£ππ,π¦π¦ = 0
πππ¦π¦ = ππ ππππππ 60Β°
ππ = 0.82 ππ/ππ2
ππ = 8.7 ππ
π¦π¦ ππ = 8.7 ππ β π¦π¦ππ = 26.9 ππ
βππ = π₯π₯ β π₯π₯ππ 2 + π¦π¦ β π¦π¦ππ 2 = 83 ππ
33Prof. Sergio B. MendesSummer 2018
Projectile Motion
Projectile Motion - PhET
from University of Colorado at Boulder
34Prof. Sergio B. MendesSummer 2018
π£π£π₯π₯ = π£π£ππ,π₯π₯ π₯π₯ ππ = π₯π₯ππ + π£π£ππ,π₯π₯ ππ
π£π£π¦π¦ = π£π£ππ,π¦π¦ π¦π¦ ππ = π¦π¦ππ + π£π£ππ,π¦π¦ ππ
π¦π¦
π₯π₯
ππ ππ = βππ οΏ½ΜοΏ½π
ππππ
ππ = βππ οΏ½ΜοΏ½π
οΏ½ΜοΏ½π
ππππ
+12πππ₯π₯ππ2+ πππ₯π₯ ππ
+ πππ¦π¦ ππβππ ππ
π£π£ππ,π₯π₯π£π£ππ,π¦π¦
π₯π₯ππ
π¦π¦ππ
+12πππ¦π¦ππ2β
12ππ ππ2
πππ₯π₯ = 0
πππ¦π¦ = βππ
35Prof. Sergio B. MendesSummer 2018
π₯π₯ ππππππ β π₯π₯ππ = π£π£ππ,π₯π₯ ππππππ = ? ?
π¦π¦ ππππππ = π¦π¦ππ + π£π£ππ,π¦π¦ππππππ β12ππ ππππππ2
π£π£ππ,π¦π¦ = 0
π¦π¦ ππππππ β π¦π¦ππ = β1.7 ππ
π£π£ππ,π₯π₯ = 31 ππ/ππ
ππππππ = ? ?
Example 3.4
ππππππ = β2π¦π¦ ππππππ β π¦π¦ππ
ππ= 0.589 πππ₯π₯ ππππππ β π₯π₯ππ = π£π£ππ,π₯π₯ ππππππ = 18 ππ
36Prof. Sergio B. MendesSummer 2018
π₯π₯ ππ = π₯π₯ππ + π£π£ππ,π₯π₯ ππ
π¦π¦ ππ = π¦π¦ππ + π£π£ππ,π¦π¦ ππ β12ππ ππ2
π¦π¦
π₯π₯
ππππ
ππ = βππ οΏ½ΜοΏ½π
οΏ½ΜοΏ½π
ππππ
π£π£ππ,π₯π₯π£π£ππ,π¦π¦
π₯π₯ππ
π¦π¦ππ
What kind of trajectory ?
π¦π¦ = π¦π¦ππ + π£π£ππ,π¦π¦π₯π₯ β π₯π₯πππ£π£ππ,π₯π₯
β12ππ
π₯π₯ β π₯π₯πππ£π£ππ,π₯π₯
2
Parabola !!
37Prof. Sergio B. MendesSummer 2018
Range of a Projectile
π¦π¦ = π¦π¦ππ + π£π£ππ,π¦π¦π₯π₯ β π₯π₯πππ£π£ππ,π₯π₯
β12ππ
π₯π₯ β π₯π₯πππ£π£ππ,π₯π₯
2
π¦π¦ = π¦π¦ππ
π₯π₯ β π₯π₯ππ =2 π£π£ππ,π₯π₯ π£π£ππ,π¦π¦
ππ=π£π£ππ2 ππππππ 2 ππππ
ππ
π¦π¦ = π¦π¦ππ
38Prof. Sergio B. MendesSummer 2018
π₯π₯ β π₯π₯ππ =π£π£ππ2 ππππππ 2 ππππ
ππ
ππ =2 π£π£ππ ππππππ ππππ
ππ
Horizontal range:
Time to return to the same height:
39Prof. Sergio B. MendesSummer 2018
Uniform Circular Velocity:circular motion with a constant magnitude of the velocity
40Prof. Sergio B. MendesSummer 2018
βππππ
=βπ£π£π£π£
π£π£1 = π£π£2ππ1 = ππ2ππ1 = ππ2π£π£1 = π£π£2
ππ =π£π£2
ππvelocity and acceleration are continuously changing (as their
directions are changing), although their magnitudes are constant
41Prof. Sergio B. MendesSummer 2018
Summary You learned to express motion quantities as vectors in one, two,
and three dimensions.
You learned that acceleration can change the velocityβs magnitude, direction, or both.
You can describe motion quantitatively when acceleration is constant.
You became familiar with projectile motion under the influence of gravity near Earthβs surface.
You became familiar with uniform circular motion.