slide1 l.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
INTRODUCTION & RECTILINEAR KINEMATICS: CONTINUOUS MOTION (Sections 12.1 - 12.2) PowerPoint Presentation
Download Presentation
INTRODUCTION & RECTILINEAR KINEMATICS: CONTINUOUS MOTION (Sections 12.1 - 12.2)

Loading in 2 Seconds...

play fullscreen
1 / 25

INTRODUCTION & RECTILINEAR KINEMATICS: CONTINUOUS MOTION (Sections 12.1 - 12.2) - PowerPoint PPT Presentation


  • 161 Views
  • Uploaded on

INTRODUCTION & RECTILINEAR KINEMATICS: CONTINUOUS MOTION (Sections 12.1 - 12.2). Today’s Objectives : Students will be able to find the kinematic quantities (position, displacement, velocity, and acceleration) of a particle traveling along a straight path. In-Class Activities :

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'INTRODUCTION & RECTILINEAR KINEMATICS: CONTINUOUS MOTION (Sections 12.1 - 12.2)' - macy


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
slide1

INTRODUCTION & RECTILINEAR KINEMATICS: CONTINUOUS MOTION (Sections 12.1 - 12.2)

Today’s Objectives:

Students will be able to find the kinematic quantities (position, displacement, velocity, and acceleration) of a particle traveling along a straight path.

In-Class Activities:

• Check homework, if any

• Reading quiz

• Applications

•Relations between s(t), v(t), and a(t) for general

rectilinear motion

•Relations between s(t), v(t),

and a(t) when acceleration is constant

• Concept quiz

• Group problem solving

• Attention quiz

slide2

Mechanics: the study of how bodies react to forces acting on them

Statics: the study of bodies in equilibrium

Dynamics:

1. Kinematics – concerned with the geometric aspects of motion

2. Kinetics - concerned with the forces causing the motion

An Overview of Mechanics

dynamics
Dynamics
  • Dynamics includes:
    • Kinematics: study of the geometry of motion. Kinematics is used to relate displacement, velocity, acceleration, and time without reference to the cause of motion.
      • Geometry of motion or Calculus of motion
    • Kinetics: study of the relations existing between the forces acting on a body, the mass of the body, and the motion of the body. Kinetics is used to predict the motion caused by given forces or to determine the forces required to produce a given motion.
      • Newtons Second Law
      • Energy Methods
      • Impulse-Momentum
slide4

Particle:

    • A body without size or shape that will be assumed to occupy a single point in space.
    • This greatly simplifies mechanics problems and when considered appropriately, gives reasonable first results.
    • Assumption: NO ROTATION
    • Caution: Can be a bit unrealistic
    • Later on we will look at Rigid Body Motion
    • Kinematics-geometry of motion
    • Kinetics-Newtons Second Law (+ some Kinematics)
  • Types of Motion
    • Rectilinear motion: position, velocity, and acceleration of a particle as it moves along a straight line.
    • Curvilinear motion: position, velocity, and acceleration of a particle as it moves along a curved line in two or three dimensions.
slide5

APPLICATIONS

The motion of large objects, such as rockets, airplanes, or cars, can often be analyzed as if they were particles.

Why?

If we measure the altitude of this rocket as a function of time, how can we determine its velocity and acceleration?

slide6

APPLICATIONS(continued)

A train travels along a straight length of track.

Can we treat the train as a particle?

If the train accelerates at a constant rate, how can we determine its position and velocity at some instant?

slide7

The displacement of the particle is defined as its change in position.

Vector form:Δ r = r’ - r

Scalar form: Δs = s’ - s

POSITION AND DISPLACEMENT

A particle travels along a straight-line path defined by the coordinate axis s.

The position of the particle at any instant, relative to the origin, O, is defined by the position vector r, or the scalar s. Scalar s can be positive or negative. Typical units for r and s are meters (m) or feet (ft).

The totaldistancetraveled by the particle, sT, is a positive scalar that represents the total length of the path over which the particle travels.

slide8

The average velocity of a particle during a time interval Δt is

vavg = Δr/Δt = Δr/Δt i

The instantaneous velocity is the time-derivative of position.

v = dr/dt

Speed is the magnitude of velocity: v = ds/dt

Average speed is the total distance traveled divided by elapsed time: (vsp)avg = sT/ Δ t

VELOCITY

Velocity is a measure of the rate of change in the position of a particle. It is a vector quantity (it has both magnitude and direction). The magnitude of the velocity is called speed, with units of m/s or ft/s.

rectilinear motion position velocity acceleration

The motion of a particle is known if the position coordinate for particle is known for every value of time t. Motion of the particle may be expressed in the form of a function, e.g.,

or in the form of a graph x vs. t.

Rectilinear Motion: Position, Velocity & Acceleration
  • Particle moving along a straight line is said to be in rectilinear motion.
  • Position coordinate of a particle is defined by positive or negative distance of particle from a fixed origin on the line.
slide10

Consider particle which occupies position P at time t and P’ at t+Δt,

Average velocity

Instantaneous velocity

  • Instantaneous velocity may be positive or negative. Magnitude of velocity is referred to as particle speed.
  • From the definition of a derivative,

e.g.,

slide11

The instantaneous acceleration is the time derivative of velocity.

Vector form: a = dv/dt

Scalar form: a = dv/dt = d2s/dt2

Acceleration can be positive (speed increasing) or negative (speed decreasing).

ACCELERATION

Acceleration is the rate of change in the velocity of a particle. It is a vector quantity. Typical units are m/s2 or ft/s2.

As the book indicates, the derivative equations for velocity and acceleration can be manipulated to get

a ds = v dv

slide12

Consider particle with velocity v at time t and v’ at t+Δt,

Instantaneous acceleration

  • Instantaneous acceleration may be:
    • positive: increasing positive velocity

or decreasing negative velocity

  • negative: decreasing positive velocity

or increasing negative velocity.

  • From the definition of a derivative,
slide13

Consider particle with motion given by

  • at t = 0, x = 0, v = 0, a = 12 m/s2
  • at t = 2 s, x = 16 m, v = vmax = 12 m/s, a = 0
  • at t = 4 s, x = xmax = 32 m, v = 0, a = -12 m/s2
  • at t = 6 s, x = 0, v = -36 m/s, a = 24 m/s2
slide14

Recall, motion of a particle is known if position is known for all time t.

  • Typically, conditions of motion are specified by the type of acceleration experienced by the particle. Determination of velocity and position requires two successive integrations.
  • Three classes of motion may be defined for:
    • acceleration given as a function of time, a = f(t)
    • - acceleration given as a function of position, a = f(x)
    • - acceleration given as a function of velocity, a = f(v)
slide15

Position:

Velocity:

v

t

v

s

s

t

ò

ò

ò

ò

ò

ò

=

=

=

dv

a

dt

or

v

dv

a

ds

ds

v

dt

v

o

v

s

s

o

o

o

o

o

SUMMARY OF KINEMATIC RELATIONS:

RECTILINEAR MOTION

•Differentiate position to get velocity and acceleration.

v = ds/dt ; a = dv/dt or a = v dv/ds

•Integrate acceleration for velocity and position.

• Note that so and vo represent the initial position and velocity of the particle at t = 0.

slide16

=

+

v

v

a

t

yields

o

c

=

+

+

s

s

v

t

(1/2)a

t

yields

o

o

c

2

=

+

v

(v

)

2a

(s

-

s

)

yields

2

2

c

o

o

CONSTANT ACCELERATION

The three kinematic equations can be integrated for the special case when accelerationis constant (a = ac) to obtain very useful equations. A common example of constant acceleration is gravity; i.e., a body freely falling toward earth. In this case, ac = g = 9.81 m/s2 = 32.2 ft/s2 downward. These equations are:

slide17

Acceleration given as a function of time, a = f(t):

  • Acceleration given as a function of position, a = f(x):
slide19

EXAMPLE

Given: A motorcyclist travels along a straight road at a speed of 27 m/s. When the brakes are applied, the motorcycle decelerates at a rate of -6t m/s2.

Find: The distance the motorcycle travels before it stops.

Plan: Establish the positive coordinate s in the direction the motorcycle is traveling. Since the acceleration is given as a function of time, integrateit once to calculate the velocity and again to calculate the position.

slide20

1) Integrate acceleration to determine the velocity.

a = dv / dt => dv = a dt =>

=> v – vo = -3t2 => v = -3t2 + vo

v

t

ò

ò

=

-

dv

(

6

t

)

dt

v

o

o

3) Now calculate the distance traveled in 3s by integrating the velocity using so = 0:

v = ds / dt => ds = v dt =>

=> s – so = -t3 + vot

=> s – 0 = (3)3 + (27)(3) => s = 54 m

s

t

ò

ò

=

-

+

ds

(

3

t

v

)

dt

2

o

s

o

o

EXAMPLE (continued)

Solution:

2) We can now determine the amount of time required for the motorcycle to stop (v = 0). Use vo = 27 m/s.

0 = -3t2 + 27 => t = 3 s

slide21

3 m/s

5 m/s

t = 2 s

t = 7 s

1. A particle moves along a horizontal path with its velocity varying with time as shown. The average acceleration of the particle is _________.

A) 0.4 m/s2 B) 0.4 m/s2

C) 1.6 m/s2 D) 1.6 m/s2

2. A particle has an initial velocity of 30 ft/s to the left. If it then passes through the same location 5 seconds later with a velocity of 50 ft/s to the right, the average velocity of the particle during the 5 s time interval is _______.

A) 10 ft/s B) 40 ft/s

C) 16 m/s D) 0 ft/s

CONCEPT QUIZ

slide22

GROUP PROBLEM SOLVING

Given: Ball A is released from rest at a height of 40 ft at the same time that ball B is thrown upward, 5 ft from the ground. The balls pass one another at a height of 20 ft.

Find: The speed at which ball B was thrown upward.

Plan: Both balls experience a constant downward acceleration of 32.2 ft/s2. Apply the formulas for constant acceleration, with ac = -32.2 ft/s2.

slide23

GROUP PROBLEM SOLVING(continued)

Solution:

1) First consider ball A. With the origin defined at the ground, ball A is released from rest ((vA)o = 0) at a height of 40 ft

((sA )o = 40 ft). Calculate the time required for ball A to drop to 20 ft (sA = 20 ft) using a position equation.

sA = (sA )o + (vA)ot + (1/2)act2

20 ft = 40 ft + (0)(t) + (1/2)(-32.2)(t2) => t = 1.115 s

2) Now consider ball B. It is throw upward from a height of 5 ft ((sB)o = 5 ft). It must reach a height of 20 ft (sB = 20 ft) at the same time ball A reaches this height (t = 1.115 s). Apply the position equation again to ball B using t = 1.115s.

sB = (sB)o + (vB)ot + (1/2) ac t2

20 ft = 5 + (vB)o(1.115) + (1/2)(-32.2)(1.115)2

=> (vB)o = 31.4 ft/s

slide24

1. A particle has an initial velocity of 3 ft/s to the left at

s0 = 0 ft. Determine its position when t = 3 s if the acceleration is 2 ft/s2 to the right.

A) 0.0 ft B) 6.0 ft

C) 18.0 ft D) 9.0 ft

ATTENTION QUIZ

2. A particle is moving with an initial velocity of v = 12 ft/s and constant acceleration of 3.78 ft/s2 in the same direction as the velocity. Determine the distance the particle has traveled when the velocity reaches 30 ft/s.

A) 50 ft B) 100 ft

C) 150 ft D) 200 ft

slide25

End of the Lecture

Let Learning Continue