Physics 207: Lecture 3, Pg 1 Physics 207, Lecture 3 Reading Assignment: For Wednesday: Read Chapter...
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Transcript of 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
Physics 207: Lecture 3, Pg 2
““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
Physics 207: Lecture 3, Pg 3
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
Physics 207: Lecture 3, Pg 4
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
Physics 207: Lecture 3, Pg 5
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)
Physics 207: Lecture 3, Pg 6
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
Physics 207: Lecture 3, Pg 7
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
Physics 207: Lecture 3, Pg 8
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
Physics 207: Lecture 3, Pg 9
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
Physics 207: Lecture 3, Pg 10
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
Physics 207: Lecture 3, Pg 11
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
Physics 207: Lecture 3, Pg 12
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
Physics 207: Lecture 3, Pg 13
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
Physics 207: Lecture 3, Pg 14
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
Physics 207: Lecture 3, Pg 15
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
Physics 207: Lecture 3, Pg 16
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
Physics 207: Lecture 3, Pg 17
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
Physics 207: Lecture 3, Pg 18
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 -)
Physics 207: Lecture 3, Pg 19
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
Physics 207: Lecture 3, Pg 20
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?