physics 207: lecture 3, pg 1 physics 207, lecture 3 reading assignment: for wednesday: read chapter...

Physics 207: Lecture 3, Pg 1

Physics 207, Lecture 3

Reading Assignment:

For Wednesday: Read Chapter 3 (carefully) through 4.4

Today (Finish Ch. 2 & start Ch. 3) Understand 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


““2D” Position, Displacement2D” Position, Displacement

time (sec) 1 2 3 4 5 6

position -2,2 -1,2 0,2 1,2 2,2 3,2 (x,y meters)

originposition vectors

x

y

displacement vectors


Position, Displacement, VelocityPosition, Displacement, Velocity

time (sec) 1 2 3 4 5 6

displacement vectors

tx

tt

xxv

if

ifx

(avg)

velocity vectors

12 23 34 45 56

Velocity always has same magnitude & length CONSTANT

2x

initial final

m/s 0/a

0

)(

tv

vvv

tvxtx

x

xxx

xi

x

y


AccelerationAcceleration Particle motion often involves non-zero acceleration

The magnitude of the velocity vector may change The direction of the velocity vector may change

(true even if the magnitude remains constant) Both may change simultaneously

E.g., a “particle” with smoothly decreasing speed

v0 v

v1

v1

01 v v v

t

va

avg

v1 v3 v5v2 v4

v v

v v

v

v0


Average & Instantaneous AccelerationAverage & Instantaneous Acceleration

Average acceleration

t

vv

tt

tvtv if

if

if

initialfinal )()(

The instantaneous acceleration is the limit of the average acceleration as ∆v/∆t approaches zero

Note: Position, velocity & acceleration are all vectors, they cannot be added directly to one another (different dimensional units)


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)

tv

a x

x

a

ti

t0 t

tf

txxx avv0


So if 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

x0

v0


Rearranging terms gives two other relationships

If constant acceleration then we also get:

)v(v2

1v

)x(x2av v

xx(avg)x

0x2

x2x

0

0


An example problem

A particle moves to the right first for 2 seconds at 1 m/s and then 4 seconds at 2 m/s.

What was the average velocity?

Two legs with constant velocity but ….

We must find the displacement (x2 –x0) And x1 = x0 + v0 (t1-t0) x2 = x1 + v1 (t2-t1) Displacement is (x2 - x1) + (x1 – x0) = v1 (t2-t1) + v0 (t1-t0) x2 –x0 = 1 m/s (2 s) + 2 m/s (4 s) = 10 m in 6 seconds or 5/3 m/s

vx

t

221

Avg

vvv


A particle starting at rest & moving along a line with constant acceleration has a displacement

whose magnitude is proportional to t2

221

0 a)( txx x

221

0 a v0

ttxx xx

1. This can be tested2. This is a potentially useful result


Speed can’t really kill but acceleration may…

“High speed motion picture camera frame: John Stapp is caught in the teeth of a massive deceleration. One might have expected that a test pilot or an astronaut candidate would be riding the sled; instead there was Stapp, a mild mannered physician and diligent physicist with a notable sense of humor. Source: US Air Force photo


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 When throwing a ball straight up, which of the following is true about its velocity true about its velocity vv and its acceleration and its acceleration aa at the highest at the highest point in its path?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 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 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 I-90 and must stop quickly so I decelerate increasingly fast until I stop. The magnitude of mystop. The magnitude of my acceleration acceleration vsvs time time is given by,is given by,

AA

AA

AA

a

t

Exercise 2 More complex Position vs. Time Graphs

• Question: My velocity vs time graph looks most 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 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. The speed of the balls when . The speed of the balls when they hit the ground arethey hit the ground are vvAA andand vvBB respectivelyrespectively..

v0

v0

BillAlice

H

vA vB


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?


See you Wednesday

Assignment: Assignment:

For Wednesday, Read through Chapter 4.4For Wednesday, Read through Chapter 4.4

physics 207: lecture 3, pg 1 physics 207, lecture 3 reading assignment: for wednesday: read chapter...

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