analysis of motion & newton’s laws. class objective zlearn and apply newton’s first, second,...

Analysis of Motion

&

Newton’s Laws

Class Objective

Learn and apply Newton’s First, Second, and Third Laws Unidirectional Multidirectional

Learn the relationship between position, velocity, and acceleration

RAT 10.1

Some Definitions (1D)

Position - location on a straight line

-3 -2 -1 0 1 2 3 4

Displacement - change in location on a straight line

-3 -2 -1 0 1 2 3 4

x

x

x = x2 - x1 = 4 - (-2) = 6


Average Velocity - rate of position change with time

TimeElapsed

ntDisplaceme

t

xvave

dt

dx

t

x

tv

0

lim

Instantaneous Velocity

Average and Instantaneous Velocity (1D)

Position

Timet1 t2

x1

x2

Slope = = Average Velocity x2 - x1

t2 - t1


Position -- a location usually described by a graphic on a map or by a coordinate system

54321-3 -2 -1

4

3

2

1

00

-1

-2

-3

-4(-2, -3)

(4, 3)


Displacement -- change in position,

where

r

4

3

2

1

1 2 3 4

00

-1-1

-2

-2

-3

-3

-4(-2, -3)

(4, 3)

51r

r

Δ

2r

12 rrrΔ


Average velocity

Instantaneous velocity

Speed - the magnitude of instantaneous velocity (scalar)

scalarTimeElapsed

vectorntDisplaceme

t

rvave

dt

rd

t

r

tv

0

lim

vspeed


Average Acceleration - rate of velocity change with time

t

v

tt

vva

12ave

12

Instantaneous Acceleration

dt

dv

t

v

0t

lima

Average and Instantaneous Acceleration (1D)

Velocity

Timet1 t2

v1

v2

Slope = = Average Acceleration

v2 - v1

t2 - t1


Average Acceleration

Instantaneous Acceleration

t

v

tt

vva

12ave

12

dt

vd

t

v

0t

lima

ExampleOne-dimensional motion

Sp

eed

,

mil e

s p

er

h

ou

r

3210

010

20

Time, hours

Paired ExerciseWhat is the distance traveled?

What is the acceleration at 1.25 hours? S

pee

d,

mil e

s p

er

h

ou

r

3210

010

20

Time, hours

For constant acceleration...

if acceleration is constant

integrating both sides

v0 is the original value at the beginning of the time interval

dt

vda

dta vd o

dtavd o

tavv oo

(Definition)

Constant Acceleration

substituting the velocity equation from the previous page

integrating both sides

yields

dt

xdv

dtv xd

dttavxd oo

dttavxd oo

2ooo ta

2

1tvxx

(Definition)

Equations of Motion (Constant Acceleration)

Velocity

Position (in terms of x)

tavv oo

2ooo ta

2

1tvxx

Multiple DirectionsEquations of motion can be written for each direction independently.

Velocity

Position

tavvo0 xxx

tavvoo yyy

2xxo ta

2

1tvxx

oo

2yyo ta

2

1tvyy

oo

Distance, Velocity, and Acceleration

Suppose a dragster has constant acceleration.

If a dragster starts from rest and accelerates to 60 mph in 10 seconds. How far did it travel?

Plot Speed vs time

60 mph

10 secondstime

spee

d

What does the area under the line represent?

(1/6 min)

(1 mi/min)

Distances….

Area = distance?

Sure: Right?

So:

xdxdtdt

dxvdt

mile12

1min

6

1

min

mile1

2

1

x

Your Turn:

RAT 10.2

Momentum

v

m

p = m v

momentum

Newton’s 1st Law:

“Every body persists in its state of rest or of uniform motion in a straight line unless it is compelled to change that state by forces impressed upon it.”

“In the absence of a net force applied to an

object, momentum stays constant.”

Newton

Holtzapple

Newton’s Second Law

The time-rate-of-change of momentum is proportional to the net force on the object.

If mass is constant...

netFdt

vmd

dt

pd

)(

netFamdt

vdm

dt

vmd

)(

Newton’s Third Law

“To every action there is always opposed an equal reaction: or, the mutual actions of two bodies upon each other are always equal and directed in contrary parts.”

Newton


Other statements

Forces always exist by the interaction of two (or more) bodies

The force on one body is equal and opposite to the force on another body

It is impossible to have a single isolated force acting in one direction

The designation of an “action force” and a “reaction force” is arbitrary because there is mutual interaction between the bodies


A Consequence

The earth and the moon orbit about a common point about 1000 miles below the surface of the earth because the earth pulls on the moon and the moon pulls on the earth.

Example: Newton’s 3rd Law

Consider a rocket with constant exhaust gas velocity:

The mass changes (obviously) as the fuel is burned and the gas is ejected.

fuelve

mv

Positive


The magnitude of the net force acting on the rocket can be determined by observing its acceleration

where m is the instantaneous mass of the rocket and dv is the instantaneous change in rocket velocity.

netFdt

vdmam


The magnitude of the net force acting on the ejected gas is

where ve is the velocity of ejected gas and dm/dt is the rate mass is ejected from the rocket.(Note: The origin of this equation will become more clear when we do Accounting for Momentum.)

dt

dmvF enet


From Newton’s 3rd Law, these two forces must be opposite and equal to each other, so:

or,

mdv

dtv

dm

dte

mdv v dme

Example: 3rd Law

Using calculus, this can be solved to yield:

where m0 is the initial mass of rocket including fuel

m m e v ve 0

/

Why Newton’s Laws?

Engineers use models to predict things such as motion, fluid flow, lift on an airplane wing, movement of neutrons in a nuclear reactor, deflection of beams or columns, etc.

Newton’s laws are widely used and a good first example of engineering models.

More on Models

Question: If I toss a piece of chalk at a sleeping student, does its path follow a parabola?

Answer: Not exactly, because air resistance affects the motion. Also, we should consider the effect of the spinning earth as it moves around the sun in an ellipse. However, for most practical work, a parabola is close enough.

analysis of motion & newton’s laws. class objective zlearn and apply newton’s first, second,...

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