Theme 2 Mechanics

Theme 2Mechanics

Module T2M1:Kinematics

1Copyright Phys 1A03, McMaster University

T2M1 – Learning Objectives• Take our day-to-day description of motion and shape

these terms into well defined quantities.

• Visually represent motion by translating our spatial observations into a quantitative picture – a graph.

• Extract information from graphs, using our kinematic definitions.

• Define a model for one dimensional motion of an object that experiences a constant acceleration.

• Extend our model to two dimensional motion to see how the vector nature of the kinematic quantities can be used with the kinematic equations.


How we define motion

Term Meaning Notation

Position Where is the object?

Displacement How far has the object moved from where it was before?

Velocity How quickly does the displacement occur?

Acceleration Is the motion getting faster or slower?


How we define motion

Term Meaning Notation

Position Where is the object?

Displacement How far has the object moved from where it was before?

Velocity How quickly does the displacement occur?

Acceleration Is the motion getting faster or slower?


Distance vs DisplacementTake a Hike!: Walk from (1) JHE to get coffee in (2) IAHS (150 m), then to (3) Health Sciences library. (200 m), then back to (1) JHE (150 m)• What total distance did you walk?

• What was your total displacement?

5

1

2

3

Copyright Phys 1A03, McMaster University

�~d = 0 m

d = 500 m

Quiz 3.4A jogger runs around a city block that is 100 m between stop signs. Starting at her apartment, she runs one and a half laps. What is her displacement from her starting point?

A. 0 m B. 100 mC. 200 m D. 500 m

start

100 m

6 6Copyright Phys 1A03, McMaster University

A jogger runs around a city block that is 100 m between stop signs. Starting at her apartment, she runs one and a half laps. What is her displacement from her starting point?

A. 0 m B. 100 mC. 200 m D. 500 m

Quiz 3.4

7

100

m

start

100 m


Quiz 3.5• When an object moves, is there anything that can be said,

in general, about distance vs. displacement?

The magnitude of the DISPLACEMENT is ___ the DISTANCE.

A. Either greater than or equal to B. Always greater thanC. Either smaller than or equal toD. Always smaller than


Quiz 3.5• When an object moves, is there anything that can be said,

in general, about distance vs. displacement?

The magnitude of the DISPLACEMENT is ___ the DISTANCE.

A. Either greater than or equal to B. Always greater thanC. Either smaller than or equal toD. Always smaller than


Plotting motion on a position-time graph

10

t (s) x (m)2 -24 106 248 2910 2712 19

0 10 20 30

2s 4s 6s 8sx(m)

Motion Diagram

t (s)

x (m)

0

10

20

4 102 6 8 12

Graph


Quiz 3.6A motion diagram shows a moving object, with equal time intervals between successive positions. Which graph represents the motion shown in the diagram below?


Quiz 3.6A motion diagram shows a moving object, with equal time intervals between successive positions. Which graph represents the motion shown in the diagram below?


Quiz 3.7• A person initially at point C on the x-axis stays there for a little

while and then strolls along the x-axis to point A, stays there for a moment and then runs to point B and remains there.

• Which graph correctlydepicts this motion?


Quiz 3.7

• A person initially at point C on the x-axis stays there for a little while and then strolls along the x-axis to point A, stays there for a moment and then runs to point B and remains there.

• Which graph correctlydepicts this motion?


Speed vs Velocity

• Vector or scalar???

• Speed = distance/time

• Velocity = displacement/time

• “How fast, and in which direction?”

15

Compare 50 km/h vs 50 km/h due northCopyright Phys 1A03, McMaster University

Speed vs VelocityYou are walking down Main St. At 1:05 pm, you are 40 m east of campus. At 1:09 pm you are 60 m west of campus. What is your average velocity over the trip?

16

Let’s define [east] as the positive direction

x1 = 40 m [East] = +40 m

x2 = 60 m [West] = �60 m

t1 = 1:05

t2 = 1:09

~v = x2�x1t2�t1

= (�60 m)�(40 m)1:09�1:05 = �100 m

4 min

= �25 m/min = �0.42 m/s What about speed?Copyright Phys 1A03, McMaster University

campus

1:05 pm

40 m

1:09 pm

-60 meastwest 12

Quiz 3.8

You go out for a jog – you travel at 8.0 km/h for 2.0 hours, 6.0 km/h for 2.0 hours and then 4.0 km/h for 1.0 hour. What is your average speed?

A. 6 km/hB. Less than 6 km/hC. Greater than 6 km/hD. Not enough information


Quiz 3.8

You go out for a jog – you travel at 8.0 km/h for 2.0 hours, 6.0 km/h for 2.0 hours and then 4.0 km/h for 1.0 hour. What is your average speed?

18

= 32 km

Answer C. Greater than 6 km/h

total time = 2 h + 2 h + 1 h = 5 h

tot. dist. = (8 km/h)(2 h) + (6 km/h)(2 h) + (4km/h)(1h)

ave. speed = (total distance)/(total time)

ave. speed = (32 km)/(5 h) = 6.4 km/h


Position-time graphs and velocity• The SLOPE of a position-time graph gives us velocity

19

t (s)

x (m)

Average velocity:


Position-time graphs and velocity• The SLOPE of a position-time graph gives us velocity

20

t (s)

x (m)

Instantaneous velocity:


Average and instantaneous velocity• The SLOPE of a position-time graph gives us velocity

21

time

position

5 10 15

2

4

0

-2

-4

When is the velocity greatest? When does the minimum velocity occur? (i.e. most –ve)When is the speed minimum?Max speed?

(t ⇠ 15 s)(t ⇠ 8 s)

(t ⇠ 4 s and ⇠ 12 s)(t ⇠ 8 s)


1D kinematics in the real world?


Sliding seal: https://twitter.com/HPS_Vanessa/status/1271903516198658069

https://twitter.com/HPS_Vanessa/status/1271903516198658069

Example• We can evaluate the instantaneous velocity of the tennis

ball while in contact with Bianca’s racquet at any point on this graph.

23

x (m)

t (s)

Ball stops

Tennis ball approaches tennis racquet

Tennis ball moves away from racquet


Example• We can evaluate the instantaneous velocity at any

point on this graph.• Can we use this to make a plot of velocity vs. time?

24

x (m)

t (s)

v (m/s)

t (s)


Acceleration• When velocity changes over a time interval

(t1à t2), we define

• Average acceleration

• What graphical interpretation will give us acceleration?

The SLOPE of a velocity-time graph gives acceleration

25

~aave =�~v

�t=

~v2 � ~v1t2 � t1


Determining acceleration from v vs. t graph

• Average acceleration

• calculated over a finite period, from t1 to t2• use the slope of the line connecting the two points

on the curve at t1 and t2.

• Instantaneous acceleration

• calculated at one value of t

• use the slope of the tangent line at that point.


Velocity-time graphs and acceleration

27

x (m)

t (s)

v (m/s)

t (s)

a (m/s2)

t (s)


Quiz 3.9

28

t (s)

a (m/s2)

A

B

C

a (m/s2)

a (m/s2)


Velocity-time graphs and acceleration

29

x (m)

t (s)

v (m/s)

t (s)

a (m/s2)

t (s)

0 slope

0 slope

0 slope

-ve slope

-ve slope

Answer: B


Tricky acceleration!

• In previous example, there are two time intervals over which the object has negative acceleration.

1. velocity is positive and acceleration is negative• the object is slowing down.

2. velocity is negative and acceleration is negative• the object is speeding up (in the negative direction).

The Big Message: Negative acceleration is NOT JUST slowing down


Quiz 3.10

• Match a given velocity graph with the corresponding acceleration graph

31

a. a-d, b-e, c-f b. a-e, b-f, c-dc. a-f, b-d, c-ed. a-e, b-d, c-f


Quiz 3.10

• Match a given velocity graph with the corresponding acceleration graph

32

a. a-d, b-e, c-f b. a-e, b-f, c-dc. a-f, b-d, c-ed. a-e, b-d, c-f


End Lecture 3B


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Theme 2 Mechanics