Linear Kinematics Chapter 3

1
Linear KinematicsChapter 3
2
Chapter Objectives

• Introduce the concept of inertia.
• Discuss the natural state of motion of objects.
• Introduce the study of kinematics.
• Describe motion.
• Lay the foundation for the study of kinetics.

3
Newtons Laws

• Every body perseveres in its state of being at
  rest or of moving uniformly straight forward,
  except insofar as it is compelled to change its
  state by force impressed
• The rate of change of momentum of a body is
  proportional to the resultant force acting on the
  body and is in the same direction
• All forces occur in pairs, and these two forces
  are equal in magnitude and opposite in direction

4
Aristotelian Physics are intuitive!

• Aristotle
• objects at rest remain at rest unless a force
  acts on them
• objects in motion did not remain in motion unless
  a force acted constantly on them

5
Aristotelian Physics are wrong!

• Galileo observed hidden forces that oppose motion
• Friction on a box sliding down an inclined plane
• Newton, who was born the year Galileo died,
  quantified this concept of motion

6
Law of Inertia

• Objects at rest tend to stay at rest.
• and
• Objects in uniform motion on a straight line
  tend to stay in motion.
• These are true unless the objects are disturbed
  by outside forces.

7
Question

• What are the units of inertia?

8
Law of Inertia contd

• Force that which causes acceleration
• Uniform motion constant velocity
• Also known as Newtons first law

9
Measuring Speed and Velocity

• Can measure speed and velocity in many ways.
• Usually measured with respect to the ground.

10
Five Ways

• Observe the object passing a series of fixed
  marks time it
• Attach a long paper tape to the object, make
  marks on the tape at a fixed frequency
• Record reflected sound pulses (Doppler)
• Use a camera
• Use infra-red light gates

11
Measure Walking Velocity

• Tools
• Tape
• Tape measure
• Stopwatch

12
Deriving Velocity from Graphs

• Displacement vs. time graph
• Independent variable plotted on the x axis (time)
• Dependent variable plotted on the y axis
  (displacement)

13
(No Transcript)
14
Generate displacement vs. time graph

• Tools
• Tape
• Tape measure
• Stopwatch

15
Deriving Velocity from Graphs contd

• For a straight line
• m slope
• b y intercept

y mx b
16
Deriving Velocity from Graphs contd

• Zero out displacement at time t0
• s displacement
• t time
• v velocity
• Then

s vt v ?s ?t
17
Split Times
Generate plot in Excel. Does velocity change?
18
Velocity and Acceleration

• Average vs. Instantaneous Velocity
• Note graphically, its all average, just over a
  smaller time span

19
Measuring Acceleration

• Average
• (final velocity initial velocity) / time
• Instantaneous?
• Slope of velocity-time curve

20
Measuring velocity and acceleration
21
Uniform Acceleration

• Acceleration rate of change of velocity
• Uniform acceleration equal changes in velocity
  over equal time intervals

22
Uniform Acceleration contd

• a acceleration
• u initial velocity
• v final velocity
• t time

v u at
23
Uniform Acceleration contd

• To find s

sut ½ at2
s½(u v)t
24
Uniform acceleration contd

• To find v
• Note Previous equations apply to translation
  with uniform acceleration only.

v2 u2 2as
25
Uniform acceleration equations
sut ½ at2
v u at
s½(u v)t
v2 u2 2as

• Note These equations apply to translation with
  uniform acceleration only.

26
Example 3.2

• A skier begins his descent of a downhill section.
  He gives himself a flying start by using his legs
  to propel himself from his starting position. As
  a result, his initial velocity is not 0 but is
  2.5 m/s as he goes past the first marker on the
  downhill section. The clock is started when he
  goes past the first marker. Given that his
  acceleration is 3 m/s2, calculate his velocity
  when he goes past the second distance marker,
  which is 10 m away from the first. How long (in
  seconds) will it take him to get up to a velocity
  of 25 m/s, assuming he maintains the acceleration
  of 3 m/s2 as long as necessary?

27

• A skier begins his descent of a downhill section.
  He gives himself a flying start by using his legs
  to propel himself from his starting position. As
  a result, his initial velocity is not 0 but is
  2.5 m/s as he goes past the first marker on the
  downhill section. The clock is started when he
  goes past the first marker. Given that his
  acceleration is 3 m/s2, calculate his velocity
  when he goes past the second distance marker,
  which is 10 m away from the first. How long (in
  seconds) will it take him to get up to a velocity
  of 25 m/s, assuming he maintains the acceleration
  of 3 m/s2 as long as necessary?

sut ½ at2
v u at
v2 u2 2as
s½(u v)t
28
Example 3.2 solution

• A skier begins his descent of a downhill section.
  He gives himself a flying start by using his legs
  to propel himself from his starting position. As
  a result, his initial velocity is not 0 but is
  2.5 m/s as he goes past the first marker on the
  downhill section. The clock is started when he
  goes past the first marker. Given that his
  acceleration is 3 m/s2, calculate his velocity
  when he goes past the second distance marker,
  which is 10 m away from the first. How long (in
  seconds) will it take him to get up to a velocity
  of 25 m/s, assuming he maintains the acceleration
  of 3 m/s2 as long as necessary?

Known u, s, a Unknown v
u2.5 m/s
s 10m
a 3 m/s2
v ?
29
Pick an equation
Known u, s, a Unknown v
u2.5 m/s
s 10m
a 3 m/s2
v ?
sut ½ at2
v u at
v2 u2 2as
s½(u v)t
30
Solve
v2 u2 2as
u2.5 m/s
s 10m
a 3 m/s2
v ?
31
Example 3.2 solution part 2

• A skier begins his descent of a downhill section.
  He gives himself a flying start by using his legs
  to propel himself from his starting position. As
  a result, his initial velocity is not 0 but is
  2.5 m/s as he goes past the first marker on the
  downhill section. The clock is started when he
  goes past the first marker. Given that his
  acceleration is 3 m/s2, calculate his velocity
  when he goes past the second distance marker,
  which is 10 m away from the first. How long (in
  seconds) will it take him to get up to a velocity
  of 25 m/s, assuming he maintains the acceleration
  of 3 m/s2 as long as necessary?

Known u, v, a Unknown t
u2.5 m/s
t ?
a 3 m/s2
v 25 m/s
32
Pick an equation
Known u, v, a Unknown t
u2.5 m/s
t ?
a 3 m/s2
v 25 m/s
sut ½ at2
v u at
v2 u2 2as
s½(u v)t
33
Solve
v u at
u2.5 m/s
t ?
a 3 m/s2
v 25 m/s
34
Acceleration due to gravity

• 9.81 m/s2
• Do you believe it?

35
Velocity vs Time Graph

• Free fall is uniform acceleration attributable to
  gravity.
• Slope of the v-vs-t graph gives acceleration.

36
(No Transcript)
37
(No Transcript)
38
Positive, Negative, or Zero!

• Americas Favorite Game Show

39
Velocity Positive, Negative, or Zero?
40
Velocity Positive, Negative, or Zero?
41
Velocity Positive, Negative, or Zero?
42
Velocity Positive, Negative, or Zero?
43
Acceleration Positive, Negative, or Zero?
44
Acceleration Positive, Negative, or Zero?
45
Acceleration Positive, Negative, or Zero?
46
Acceleration Positive, Negative, or Zero?
Note acceleration is the second derivative of
displacement
47
Frames of Reference

• Usually fixed with respect to surfaces of the
  Earth
• Can also be moving with respect to the Earth
• Relative Velocity
• Velocity WRT

48

• Man WRT train
• Woman WRT man

• Describe velocity of
• Train
• Man on train
• Woman on platform

49

• Man WRT ball?

• Describe velocity of
• Ball WRT woman
• Ball WRT man

50

• Consider relative velocities in throwing events
• Javelin thrower is not standing still at release
• Add velocities to get javelin WRT ground

51
Projectiles

• Ignoring friction, projectiles move with uniform
  acceleration
• Examples of projectiles
• Shot putt
• High jumper
• Long jumper
• Soccer ball
• Tennis ball

52
Projectiles contd

• Horizontal and vertical components of velocity in
  projectiles
• Horizontal component is constant in the absence
  of friction

Vx Vcos? X Vxt
53
Projectiles contd

• Vertical component
• At top of projectile
• max height h, time to max height tup

Vy Vsin?
Vy 0 at t tup v u at 0 vsin? - g tup
tup vsin? / g
54
Projectiles contd

• Define positive and negative direction.
• Up
• Down -
• Keep values constant

55
Horizontal Range
See pg. 67-68 for flight time calculation

• Horizontal range of projectile
• If Landing height Release height,

56
Optimum projection angle
57
Release Height Not at Landing Height
dxh

• d is Descent distance
• h is maximum height
• x is projection height minus landing height
• x is positive if the projection level is higher
  than the landing level.
• x is negative if the projection level is lower
  than the landing level.

58
Parabolic Path

• A projectiles path in two dimensions forms a
  parabola.

Solve for t
59
Parabolic Path contd

• Equation of parabola in the xy plane

60
Proof
61

• Ball projected at 45º angle
• Path of ball illuminated stroboscopically at a
  frequency of 28 Hz
• Projection velocity approx. 4 m/s

62
Note

• Horizontal velocity is constant once released
• 0.1 m across each flash

63
Note

• Vertical velocity varies at a constant rate
• Trajectory is symmetric

64
Summary

• The natural motion of a body is constant
  velocity in a straight line
• However, we more often see objects that are
  stationary or moving with a constant acceleration
• Do you actually believe Aristotle or Galileo and
  Newton?