Physics 207: Lecture 3, Pg 1 Physics 207, Lecture 3 Assignment: 1.For Thursday: Read Chapter 3...

Physics 207: Lecture 3, Pg 1

Physics 207, Lecture 3

Assignment:

1. For Thursday: Read Chapter 3 (carefully) through 4.4

2. Homework Set 2 available at noon

Goals (finish Chap. 2 & 3) Understand the relationships between position, velocity

& acceleration in systems with 1-dimensional motion and non-zero acceleration (usually constant)

Solve problems with zero and constant acceleration (including free-fall and motion on an incline)

Use Cartesian and polar coordinate systems

Perform vector algebra


Position, velocity & acceleration for motion along a line

If the position x is known as a function of time, then we can find both the instantaneous velocity vx and instantaneous acceleration ax as a function of time!

t

vx

t

x

ax

t

2

2va

dtxd

dtd x

x

dtdx

x v

] offunction a is [ )( txtxx


Going the other way…. Particle motion with constant acceleration

The magnitude of the velocity vector changes A particle with smoothly decreasing speed:

vv11vv00 vv33 vv55vv22 vv44

aa aa aa aa aa aa

v

t0

vf = vi + a t

= vi + a (tf - ti )

a t = area under curve = v (an integral)

t

va

avg

t

va

t

0inst lim

a

ti

t0 t

tf


And given a constant acceleration we can integrate to get explicit v and a

x

ax

vx

t

t

t

txxx avv0

221

0 a v0

ttxx xx

consta x

2

2va

dtxd

dtd x

x

dt

dxx v

] offunction a is [ )( txtxx


Rearranging terms gives two other relationships

For constant acceleration:

From which we can show (caveat: a constant acceleration)

)v(v21

v

)x(x2av v

xx(avg)x

0x2

x2x

0

0

txxx avv0

221

0 a v0

ttxx xx

consta x

ax

t


Free Fall

When any object is let go it falls toward the ground !! The force that causes the objects to fall is called gravity.

This acceleration on the Earth’s surface, caused by gravity, is typically written as “little” g

Any object, be it a baseball or an elephant, experiences the same acceleration (g) when it is dropped, thrown, spit, or hurled, i.e. g is a constant.

221

0 g v)(0

ttyty y ga -y


Gravity facts:

g does not depend on the nature of the material !

Galileo (1564-1642) figured this out without fancy clocks & rulers!

Feather & penny behave just the same in

vacuum

Nominally, g = 9.81 m/s2 At the equator g = 9.78 m/s2

At the North pole g = 9.83 m/s2


When throwing a ball straight up, which of the following is true about its velocity v and its acceleration a at the highest point in its path?

A. Both v = 0 and a = 0

B. v 0, but a = 0

C. v = 0, but a 0

D. None of the above

y

Exercise 1Motion in One Dimension


In driving from Madison to Chicago, initially my speed is at a constant 65 mph. After some time, I see an accident ahead of me on I-90 and must stop quickly so I decelerate increasingly “fast” until I stop. The magnitude of my acceleration vs time is given by,

AA

AA

AA

a

t

Exercise 2 More complex Position vs. Time Graphs

• Question: My velocity vs time graph looks like which of the following ?

v

t

v

v


Exercise 3 1D Freefall

A. vA < vB

B. vA = vB

C. vA > vB

Alice and Bill are standing at the top of a cliff of height H. Both throw a ball with initial speed v0, Alice straight down and Bill straight up.

The speed of the balls when they hit the ground are vA and vB respectively.. ( (Neglect air resistance.)

vv00

vv00

BillAliceAlice

HH

vvAA vvBB


Exercise 3 1D Freefall : Graphical solution

Alice and Bill are standing at the top of a cliff of heightAlice and Bill are standing at the top of a cliff of height HH. Both throw a ball with initial speed. Both throw a ball with initial speed vv00, Alice straight, Alice straight downdown and Bill straightand Bill straight upup. .

vx t

cliff back

at

cliff

turnaround

point

groundground

v0

-v0

vground

v= -g t

identical displacements

(one + and one -)


The graph at right shows the The graph at right shows the yy velocity versus velocity versus timetime graph for a graph for a ball. Gravity is acting downward ball. Gravity is acting downward in the in the -y-y direction and the direction and the x-x-axis axis is along the horizontal. is along the horizontal.

Which explanation Which explanation best fitsbest fits the the motion of the ball as shown by motion of the ball as shown by the velocity-time graph below?the velocity-time graph below?

A. The ball is falling straight down, is caught, and is then thrown straight down with greater velocity.

B. The ball is rolling horizontally, stops, and then continues rolling.

C. The ball is rising straight up, hits the ceiling, bounces, and then falls straight down.

D. The ball is falling straight down, hits the floor, and then bounces straight up.

E. The ball is rising straight up, is caught and held for awhile, and then is thrown straight down.

Home Exercise,1D Freefall


Problem Solution Method:

Five Steps:

1) Focus the Problem- draw a picture – what are we asking for?

2) Describe the physics- what physics ideas are applicable

- what are the relevant variables known and unknown

3) Plan the solution- what are the relevant physics equations

4) Execute the plan- solve in terms of variables

- solve in terms of numbers

5) Evaluate the answer- are the dimensions and units correct?

- do the numbers make sense?


Example of a 1D motion problem A cart is initially traveling East at a constant speed of 20 m/s.

When it is halfway (in distance!) to its destination its speed suddenly increases and thereafter remains constant. All told the cart spends a total of 10 s in transit with an average speed of 25 m/s.

What is the speed of the cart during the 2nd half of the trip? Dynamical relationships (only if constant acceleration):

txxx avv0

221

0 a v0

ttxx xx

consta x)v(v

21

v

)x(x2av v

xx(avg)x

0x2

x2x

0

0

) timetotal(

)ntdisplaceme( velocityaveragev

tx

And


Drawing the picture

Plus the average velocity Knowns:

x0 = 0 m

t0 = 0 s

v0 = 20 m/s

t2 = 10 s

vavg = 25 m/s

relationship between x1 and x2

Four unknowns x1 v1 t1 & x2 and must find v1 in terms of knowns

t2t0 t1

v1 ( > v0 )

a1=0 m/s2

v0

a0=0 m/s2

x0 x1 x2

02

02vttxx

tx

0

vv xx txx x v

00

0a x


Using

Four unknowns Four relationships Focus on

t2t0 t1

v1 ( > v0 )

a1=0 m/s2

v0

a0=0 m/s2

x0 x1 x2

)(

v

0221

1

02

02

xxx

ttxx

tx

txx x v0

0

)( v 01001 ttxx )( v 12112 ttxx


Using

t2t0 t1

v1 ( > v0 )

a1=0 m/s2

v0

a0=0 m/s2

x0 x1 x2

s )4/15(m 125

s )4/2510(

m 125s 10

s 25/4 s 20/125/

m 125 m 125)(

m 250 s 1025vv

12

1222

111

120221

1

sm

2

02

02

ttxx

vt

vxt

xxxxx

txttxx

tx

txx x v0

0


More formally 4 eqns., 4 unknowns

Eliminate unknowns

first x1

next t1

then x2

t2t0 t1

v1 ( > v0 )

a1=0 m/s2

v0

a0=0 m/s2

x0 x1 x2

221

1 xx

0 0 00 tx

101 v tx )( v 12112 ttxx

)( v 121221

2 ttxx 2

3

2

)( v0

221

v21221 xtx 1&2

1

2

2vt

x4

4 )( vv0

221

vv

21221 ttt )2( vv

0vv

1 Mult. by 2/ t2

1 10221 v tx


Fini

Plus the average velocity Given:

v0 = 20 m/s

t2 = 10 s

vavg = 25 m/s

t2t0 t1

v1 ( > v0 )

a1=0 m/s2

v0

a0=0 m/s2

x0 x1 x2

m/s 3.33

v

v

)2( vv

15m/s 005

m/s 25m/s 20 2m/s 20 m/s 25

1

vv2 vv

1

vv

1

0

0

0


Tips: Read !

Before you start work on a problem, read the problem statement thoroughly. Make sure you understand what information is given, what is asked for, and the meaning of all the terms used in stating the problem.

Watch your units (dimensional analysis) !Always check the units of your answer, and carry the

units along with your numbers during the calculation.

Ask questions !


Coordinate Systems and vectorsCoordinate Systems and vectors

In 1 dimension, only 1 kind of system, Linear Coordinates (x) +/-

In 2 dimensions there are two commonly used systems,Cartesian Coordinates (x,y)Circular Coordinates (r,)

In 3 dimensions there are three commonly used systems,Cartesian Coordinates (x,y,z)Cylindrical Coordinates (r,,z)Spherical Coordinates (r,)


See you Thursday

Assignment: Assignment:

For Monday, Read through Chapter 4.4For Monday, Read through Chapter 4.4

Mastering Physics Problem Set 2Mastering Physics Problem Set 2

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Physics 207: Lecture 3, Pg 1 Physics 207, Lecture 3 Assignment: 1.For Thursday: Read Chapter 3...

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