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1Prof. Sergio B. MendesSummer 2018
Mechanics
Chapter 2 of Essential University Physics, Richard Wolfson, 3rd Edition
2Prof. Sergio B. MendesSummer 2018
Mechanics
DynamicsKinematics• How something moves ?? • Why something moves ??
• Geometrical description
• Mathematics
• Physical cause
• Physics
3Prof. Sergio B. MendesSummer 2018
Kinematics
• Object in motion: a small particle (a point)
• 1D: along a straight direction
𝑥𝑥, 𝑡𝑡 𝑥𝑥×
𝑥𝑥 = 0
𝓞𝓞
4Prof. Sergio B. MendesSummer 2018
Average Velocity
�̅�𝑣 ≡𝑥𝑥2 − 𝑥𝑥1𝑡𝑡2 − 𝑡𝑡1
Between two well-defined events.
𝑥𝑥1, 𝑡𝑡1 𝑥𝑥2, 𝑡𝑡2
Event “1” Event “2”
Average between WHAT ??
𝑥𝑥×
𝑥𝑥 = 0
𝓞𝓞
Average Velocity
5Prof. Sergio B. MendesSummer 2018
Examples
6Prof. Sergio B. MendesSummer 2018
Displaying the Two Events in a Plot
𝑥𝑥
𝑡𝑡
𝑥𝑥1, 𝑡𝑡1Event “1”
𝑥𝑥2, 𝑡𝑡2Event “2”
𝑡𝑡2𝑡𝑡1
𝑥𝑥2
𝑥𝑥1𝑡𝑡2 − 𝑡𝑡1
𝑥𝑥2 − 𝑥𝑥1
𝜃𝜃
�̅�𝑣 ≡𝑥𝑥2 − 𝑥𝑥1𝑡𝑡2 − 𝑡𝑡1
= 𝑡𝑡𝑡𝑡𝑡𝑡 𝜃𝜃
7Prof. Sergio B. MendesSummer 2018
Making a Plot to Display the Particle Motion: 𝑥𝑥 𝑡𝑡
𝑡𝑡
𝑥𝑥1, 𝑡𝑡1Event “1”
𝑥𝑥2, 𝑡𝑡2Event “2”
𝑡𝑡2𝑡𝑡1
𝑥𝑥2
𝑥𝑥1𝑡𝑡2 − 𝑡𝑡1
𝑥𝑥2 − 𝑥𝑥1
�̅�𝑣 = �̅�𝑣 = �̅�𝑣 = �̅�𝑣Regardless of the different types of motion:
𝑥𝑥 𝑡𝑡
8Prof. Sergio B. MendesSummer 2018
Instantaneous Velocity
𝑡𝑡
𝑥𝑥1, 𝑡𝑡1Event “1”
Event “2”
𝑡𝑡1
𝑥𝑥1𝜃𝜃1
𝑥𝑥2 = 𝑥𝑥1 + ∆𝑥𝑥, 𝑡𝑡2 = 𝑡𝑡1 + ∆𝑡𝑡
= lim∆𝑡𝑡→0
∆𝑥𝑥∆𝑡𝑡
𝑣𝑣 𝑡𝑡1 ≡ lim∆𝑡𝑡→0
𝑥𝑥2 − 𝑥𝑥1𝑡𝑡2 − 𝑡𝑡1
= 𝑡𝑡𝑡𝑡𝑡𝑡 𝜃𝜃1 =𝑑𝑑𝑥𝑥 𝑡𝑡 = 𝑡𝑡1
𝑑𝑑𝑡𝑡
𝑡𝑡2 − 𝑡𝑡1 = ∆𝑡𝑡
𝑥𝑥2 − 𝑥𝑥1 = ∆𝑥𝑥
𝑥𝑥 𝑡𝑡
9Prof. Sergio B. MendesSummer 2018
Instantaneous Velocity𝑥𝑥 𝑡𝑡
𝑡𝑡
𝑣𝑣 𝑡𝑡 = 𝑡𝑡𝑡𝑡𝑡𝑡 𝜃𝜃 𝑡𝑡 =𝑑𝑑𝑥𝑥 𝑡𝑡𝑑𝑑𝑡𝑡
𝜃𝜃 𝑡𝑡
10Prof. Sergio B. MendesSummer 2018
Evaluating Instantaneous Velocity
𝑡𝑡
𝑥𝑥1, 𝑡𝑡1Event “1”
𝑥𝑥2, 𝑡𝑡2Event “2”
𝑡𝑡2𝑡𝑡1
𝑥𝑥1
𝑥𝑥 𝑡𝑡
11Prof. Sergio B. MendesSummer 2018
Acceleration
12Prof. Sergio B. MendesSummer 2018
𝑡𝑡
𝑣𝑣1, 𝑡𝑡1
𝑣𝑣2, 𝑡𝑡2
𝑡𝑡2𝑡𝑡1
𝑣𝑣2
𝑣𝑣1𝑡𝑡2 − 𝑡𝑡1
𝑣𝑣2 − 𝑣𝑣1
𝜙𝜙
= 𝑡𝑡𝑡𝑡𝑡𝑡 𝜙𝜙
Average Acceleration
�𝑡𝑡 ≡𝑣𝑣2 − 𝑣𝑣1𝑡𝑡2 − 𝑡𝑡1
𝑣𝑣 𝑡𝑡
13Prof. Sergio B. MendesSummer 2018
Instantaneous Acceleration𝑣𝑣
𝑡𝑡
𝑣𝑣1, 𝑡𝑡1
𝑡𝑡1
𝑣𝑣1𝜙𝜙1
𝑣𝑣2 = 𝑣𝑣1 + ∆𝑣𝑣, 𝑡𝑡2 = 𝑡𝑡1 + ∆𝑡𝑡
= lim∆𝑡𝑡→0
∆𝑣𝑣∆𝑡𝑡
𝑡𝑡 𝑡𝑡1 ≡ lim∆𝑡𝑡→0
𝑣𝑣2 − 𝑣𝑣1𝑡𝑡2 − 𝑡𝑡1
= 𝑡𝑡𝑡𝑡𝑡𝑡 𝜙𝜙1 =𝑑𝑑𝑣𝑣 𝑡𝑡 = 𝑡𝑡1
𝑑𝑑𝑡𝑡
𝑡𝑡2 − 𝑡𝑡1 = ∆𝑡𝑡
𝑣𝑣2 − 𝑣𝑣1 = ∆𝑣𝑣
14Prof. Sergio B. MendesSummer 2018
Instantaneous Acceleration𝑣𝑣 𝑡𝑡
𝑡𝑡
𝑡𝑡 𝑡𝑡 = 𝑡𝑡𝑡𝑡𝑡𝑡 𝜙𝜙 𝑡𝑡 =𝑑𝑑𝑣𝑣 𝑡𝑡𝑑𝑑𝑡𝑡
𝜙𝜙 𝑡𝑡
15Prof. Sergio B. MendesSummer 2018
𝑣𝑣 𝑡𝑡 =𝑑𝑑𝑥𝑥 𝑡𝑡𝑑𝑑𝑡𝑡
𝑥𝑥 𝑡𝑡
𝑡𝑡 𝑡𝑡 =𝑑𝑑𝑣𝑣 𝑡𝑡𝑑𝑑𝑡𝑡 =
𝑑𝑑𝑑𝑑𝑡𝑡
𝑑𝑑𝑥𝑥 𝑡𝑡𝑑𝑑𝑡𝑡
=𝑑𝑑2𝑥𝑥 𝑡𝑡𝑑𝑑𝑡𝑡2
16Prof. Sergio B. MendesSummer 2018
Constant Acceleration
𝑡𝑡 𝑡𝑡 = 𝑡𝑡
�𝑡𝑡 = 𝑡𝑡 =𝑣𝑣 𝑡𝑡 − 𝑣𝑣𝑜𝑜𝑡𝑡 − 0
�𝑡𝑡 = 𝑡𝑡
𝑣𝑣 𝑡𝑡 = 𝑣𝑣𝑜𝑜 + 𝑡𝑡 𝑡𝑡
17Prof. Sergio B. MendesSummer 2018
𝑣𝑣 𝑡𝑡 = 𝑣𝑣𝑜𝑜 + 𝑡𝑡 𝑡𝑡
𝑣𝑣 𝑡𝑡
𝑡𝑡0
𝑣𝑣𝑜𝑜
�̅�𝑣 = 𝑣𝑣𝑜𝑜 +12𝑡𝑡 𝑡𝑡 =
𝑥𝑥 𝑡𝑡 − 𝑥𝑥𝑜𝑜𝑡𝑡 − 0
𝑣𝑣𝑜𝑜 + 𝑡𝑡 𝑡𝑡
𝑥𝑥 𝑡𝑡 = 𝑥𝑥𝑜𝑜 + 𝑣𝑣𝑜𝑜 𝑡𝑡 +12𝑡𝑡 𝑡𝑡 2
18Prof. Sergio B. MendesSummer 2018
𝑥𝑥 𝑡𝑡 = 𝑥𝑥𝑜𝑜 + 𝑣𝑣𝑜𝑜 𝑡𝑡 +12𝑡𝑡 𝑡𝑡 2
𝑣𝑣𝑜𝑜 = 0
19Prof. Sergio B. MendesSummer 2018
𝑥𝑥 𝑡𝑡 = 𝑥𝑥𝑜𝑜 + 𝑣𝑣𝑜𝑜 𝑡𝑡 +12𝑡𝑡 𝑡𝑡 2
𝑣𝑣 𝑡𝑡 = 𝑣𝑣𝑜𝑜 + 𝑡𝑡 𝑡𝑡
𝑡𝑡 𝑡𝑡 = 𝑡𝑡
Constant Acceleration
𝑣𝑣2 𝑡𝑡 = 𝑣𝑣𝑜𝑜2 + 2 𝑡𝑡 𝑥𝑥 𝑡𝑡 − 𝑥𝑥𝑜𝑜
20Prof. Sergio B. MendesSummer 2018
0 = 𝑣𝑣𝑜𝑜 + 𝑡𝑡 𝑡𝑡𝑡
𝑡𝑡𝑡 = −𝑣𝑣𝑜𝑜𝑡𝑡
When does the velocity go to zero ?
𝑣𝑣 𝑡𝑡
𝑡𝑡
𝑣𝑣𝑜𝑜
𝑡𝑡𝑡
𝑣𝑣 𝑡𝑡 = 𝑣𝑣𝑜𝑜 + 𝑡𝑡 𝑡𝑡
21Prof. Sergio B. MendesSummer 2018
𝑡𝑡𝑡 = −𝑣𝑣𝑜𝑜𝑡𝑡
𝑥𝑥 𝑡𝑡𝑡 = 𝑥𝑥𝑜𝑜 + 𝑣𝑣𝑜𝑜 𝑡𝑡𝑡 +12𝑡𝑡 𝑡𝑡𝑡2
= 𝑥𝑥𝑜𝑜 −𝑣𝑣𝑜𝑜2
2 𝑡𝑡
𝑥𝑥 𝑡𝑡
𝑡𝑡𝑡𝑡𝑡 = −𝑣𝑣𝑜𝑜𝑡𝑡
𝑥𝑥 𝑡𝑡𝑡 = 𝑥𝑥𝑜𝑜 −𝑣𝑣𝑜𝑜2
2 𝑡𝑡
𝑥𝑥𝑜𝑜
At the time the velocity goes to zero, then
22Prof. Sergio B. MendesSummer 2018
Constant Acceleration due to Gravity𝑥𝑥 → 𝑦𝑦
𝑦𝑦 𝑡𝑡 = 𝑦𝑦𝑜𝑜 + 𝑣𝑣𝑜𝑜 𝑡𝑡 −12𝑔𝑔 𝑡𝑡 2
𝑣𝑣 𝑡𝑡 = 𝑣𝑣𝑜𝑜 − 𝑔𝑔 𝑡𝑡
𝑣𝑣2 𝑡𝑡 = 𝑣𝑣𝑜𝑜2 − 2 𝑔𝑔 𝑦𝑦 𝑡𝑡 − 𝑦𝑦𝑜𝑜
𝑦𝑦
𝑡𝑡 𝑡𝑡 = −𝑔𝑔 𝑔𝑔 ≅ 9.8 𝑚𝑚/𝑠𝑠2
23Prof. Sergio B. MendesSummer 2018
𝑡𝑡𝑡 =𝑣𝑣𝑜𝑜𝑔𝑔 𝑦𝑦 𝑡𝑡𝑡 = 𝑦𝑦𝑜𝑜 +
𝑣𝑣𝑜𝑜2
2 𝑔𝑔
𝑡𝑡𝑡
𝑦𝑦 𝑡𝑡𝑡
𝑦𝑦 𝑡𝑡 = 𝑦𝑦𝑜𝑜 + 𝑣𝑣𝑜𝑜 𝑡𝑡 −12𝑔𝑔 𝑡𝑡 2
𝑣𝑣 𝑡𝑡 = 𝑣𝑣𝑜𝑜 − 𝑔𝑔 𝑡𝑡
𝑡𝑡 𝑡𝑡 = −𝑔𝑔
𝑡𝑡𝑡
𝑦𝑦𝑜𝑜
24Prof. Sergio B. MendesSummer 2018
You toss a ball straight up at 7.3 m/s; it leaves your hand at 1.5 m above the floor.
a) Find its maximum height.
b) Find when it hits the floor.
a) At the maximum height the ball is instantaneously at rest: v = 0
𝑦𝑦𝑜𝑜 = 1.5 𝑚𝑚𝑣𝑣𝑜𝑜 = 7.3 𝑚𝑚/𝑠𝑠
𝑣𝑣2 = 𝑣𝑣𝑜𝑜2 − 2 𝑔𝑔 𝑦𝑦 − 𝑦𝑦𝑜𝑜
What do we know?
𝑔𝑔 ≅ 9.8 𝑚𝑚/𝑠𝑠2
𝑦𝑦 = 𝑦𝑦𝑜𝑜 +𝑣𝑣𝑜𝑜2
2 𝑔𝑔= 4.2 𝑚𝑚
b) When it hits the floor: y = 0
𝑦𝑦 = 𝑦𝑦𝑜𝑜 + 𝑣𝑣𝑜𝑜 𝑡𝑡 −12𝑔𝑔 𝑡𝑡 2 𝑡𝑡 =
−𝑣𝑣𝑜𝑜 ± 𝑣𝑣𝑜𝑜2 + 2𝑦𝑦𝑜𝑜 𝑔𝑔−𝑔𝑔
−0.18 𝑠𝑠
+1.7 𝑠𝑠
25Prof. Sergio B. MendesSummer 2018
𝑑𝑑𝑥𝑥 𝑡𝑡𝑑𝑑𝑡𝑡
= 𝑣𝑣 𝑡𝑡
𝑥𝑥 𝑡𝑡
𝑑𝑑𝑣𝑣 𝑡𝑡𝑑𝑑𝑡𝑡
= 𝑡𝑡 𝑡𝑡
�0
𝑡𝑡𝑣𝑣 𝑡𝑡 𝑑𝑑𝑡𝑡 = 𝑥𝑥 𝑡𝑡 − 𝑥𝑥𝑜𝑜
�0
𝑡𝑡𝑡𝑡 𝑡𝑡 𝑑𝑑𝑡𝑡 = 𝑣𝑣 𝑡𝑡 − 𝑣𝑣𝑜𝑜
Big Picture of Chapter 2:Kinematics in 1D
26Prof. Sergio B. MendesSummer 2018
𝑡𝑡 𝑡𝑡 = 𝑡𝑡
𝑥𝑥 𝑡𝑡 − 𝑥𝑥𝑜𝑜 = 𝑣𝑣𝑜𝑜 𝑡𝑡 +12𝑡𝑡 𝑡𝑡 2 = �
0
𝑡𝑡𝑣𝑣 𝑡𝑡 𝑑𝑑𝑡𝑡
�0
𝑡𝑡𝑡𝑡 𝑡𝑡 𝑑𝑑𝑡𝑡 = 𝑡𝑡 𝑡𝑡 = 𝑣𝑣 𝑡𝑡 − 𝑣𝑣𝑜𝑜
For example, in the case ofConstant Acceleration
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