Physics 207, Lecture 3, Sept. 10
Download
Report
Transcript Physics 207, Lecture 3, Sept. 10
Physics 207, Lecture 3, Sept. 10
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
Assignment:
1. For Monday: Read Chapter 4
2. Homework Set 2 (due Wednesday 9/17)
Physics 207: Lecture 3, Pg 1
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!
x
x x(t ) [ x is a function of t ]
dx
vx
dt
dv x d 2 x
ax
2
dt
dt
t
vx
t
ax
t
Physics 207: Lecture 3, Pg 2
Position, displacement, velocity & acceleration
All are vectors and so vector algebra is a must !
These cannot be used interchangeably (different units!)
(e.g., position vectors cannot be added directly to velocity
vectors)
But we can determined directions
“Change in the position” vector gives the direction of
the velocity vector
v
“Change in the velocity” vector gives the direction of
the acceleration vector
a
Given x(t) vx(t) ax (t)
Given ax (t) vx (t) x(t)
Physics 207: Lecture 3, Pg 3
And given a constant acceleration we
can integrate to get explicit vx and ax
x x(t ) [ x is a function of t ]
dx
vx
dt
dv x d 2 x
ax
2
dt
dt
x0
t
vx
x x0 v x t a x t
0
x
1
2
0
2
t
ax
v x v x a x t
0
a x const
t
Physics 207: Lecture 3, Pg 4
Exploiting
( x x0 ) t
2
A “biology” experiment
Hypothesis: Older people have slower reaction
times
Distance accentuates the impact of time differences
Equipment: Ruler and four volunteers
Older student
Younger student
Record keeper
Statistician
Expt. require multiple trials to reduce statistical
errors.
Physics 207: Lecture 3, Pg 5
Rearranging terms gives two other relationships
For constant acceleration:
x x0 v x t 12 a x t 2
0
v x v x a x t
0
a x const
From which we can show (caveat: a constant acceleration)
v 2x v x2 2a x (x x 0 )
0
v x (avg)
1
(v x v x )
2
ax
0
t
Physics 207: Lecture 3, Pg 6
Acceleration
Changes in a particle’s motion often involve 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
a
v0
v1
v2
v3
v4
v5
a
a
a
a
a
a
t
t
a t = area under curve = v
v
0
v(t)=v0 + a t
t
Physics 207: Lecture 3, Pg 7
Acceleration has its limits
“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 8
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.
Physics 207: Lecture 3, Pg 9
Exercise 1
Motion in One Dimension
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.
B.
C.
D.
Both v = 0 and a = 0
v 0, but a = 0
v = 0, but a 0
None of the above
y
Physics 207: Lecture 3, Pg 10
Exercise 1
Motion in One Dimension
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.
B.
C.
D.
Both v = 0 and a = 0
v 0, but a = 0
v = 0, but a 0
None of the above
y
Physics 207: Lecture 3, Pg 11
Exercise 2
More complex Position vs. Time Graphs
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,
•
t
Question: My velocity vs time graph looks
like which of the following ?
a
A.
v
t
B.
C.
v
v
Physics 207: Lecture 3, Pg 12
Exercise 2
More complex Position vs. Time Graphs
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,
•
t
Question: My velocity vs time graph looks
like which of the following ?
a
A.
v
t
B.
C.
v
v
Physics 207: Lecture 3, Pg 13
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 14
Gravity Map of the Earth
(relief exaggerated)
A person off the
coast of India would
weigh 1% less
than at most other
places on earth.
Physics 207: Lecture 3, Pg 15
Gravity map of the US
Red: Areas of stronger local g
Blue: Areas of weaker local g
Due to density variations of the Earth’s crust and mantle
Physics 207: Lecture 3, Pg 16
Exercise 3 1D Freefall
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.
A.
vA < vB
Alice
B.
vA = v B
v0
Bill
v0
C.
vA > vB
H
vA
vB
Physics 207: Lecture 3, Pg 17
Exercise 3 1D Freefall
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.
A.
vA < vB
Alice
B.
vA = v B
v0
Bill
v0
C.
vA > vB
H
vA
vB
Physics 207: Lecture 3, Pg 18
Problem Solution Method:
Five Steps:
1)
Focus the Problem
-
2)
Describe the physics
-
3)
what are the relevant physics equations
Execute the plan
-
5)
what physics ideas are applicable
what are the relevant variables known and unknown
Plan the solution
-
4)
draw a picture – what are we asking for?
solve in terms of variables
solve in terms of numbers
Evaluate the answer
-
are the dimensions and units correct?
do the numbers make sense?
Physics 207: Lecture 3, Pg 20
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:
x x0 v x t a x t
0
v x v x a x t
0
a x const
And
1
2
2
v 2x v x2 2a x (x x 0 )
0
v x (avg)
1
(v x v x )
2
0
x(displaceme nt )
vaverage velocity
t ( total time )
Physics 207: Lecture 3, Pg 21
The picture
x0
t0
v0
v1 ( > v0 )
a0=0 m/s2
a1=0 m/s2
x1 t1
x
x x2 x0
v
t t2 t0
t2
2
Plus the average velocity
Knowns:
x0 = 0 m
t0 = 0 s
0
x
v0 = 20 m/s
vx vx
t2 = 10 s
ax 0
vavg = 25 m/s
relationship between x1 and x2
Four unknowns x1 v1 t1 & x2 and must find v1 in terms of knowns
x x v t
0
0
Physics 207: Lecture 3, Pg 22
x x0 v x t
Using
0
x0
v0
v1 ( > v0 )
a0=0 m/s2
a1=0 m/s2
t0
x1 x0 v0 (t1 t0 )
Four
unknowns
Four
relationships
x1 t1
x
x2 x1 v1 (t2 t1 )
t2
2
x x2 x0
v
t t2 t0
x1 ( x2 x0 )
1
2
Physics 207: Lecture 3, Pg 23
x0 0
Using
x0
v0
v1 ( > v0 )
a0=0 m/s2
a1=0 m/s2
t0
x1 t1
1
x1 v0 t1
Eliminate
unknowns
first t1
3
next x1
t0 0
1&2
3
2
x
x2 x1 v1 (t2 t1 )
x1 x2
1
2
t2
2
x2
v
t2
4
x2 x1 v1 (t2 )
x1
v0
x2
x v1 (t2 )
1
2 2
x2
v0
1
2
Physics 207: Lecture 3, Pg 24
Now Algebra and Relationship 4
x0
4
v1 ( > v0 )
a0=0 m/s2
a1=0 m/s2
t0
x1 t1
Algebra to simplify
x2
1
2
v0
1
2
x2 v1 (t2 )
x2
v0
1
2
x2 v1 (t2 )
x2
v0
x2
v
t2
x
t2
2
1
2
vt2 v1 (2t2 )
v t2
v0
v1 (
v v0
2 v0 v
)
v v1 (2 vv )
0
Physics 207: Lecture 3, Pg 25
Fini
x0
v0
v1 ( > v0 )
a0=0 m/s2
a1=0 m/s2
t0
Plus the average velocity
Given:
v0 = 20 m/s
t2 = 10 s
vavg = 25 m/s
x1 t1
x
2
v1
v v0
2 v0 v
v1
25 m/s 20 m/s
2 20 m/s 25 m/s
t2
500 m/s
15
33.3 m/s
Physics 207: Lecture 3, Pg 26
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 !
Physics 207: Lecture 3, Pg 27
See you Monday
(Chapter 3 on Monday….)
Assignment:
For Monday, Read Chapter 4
Mastering Physics Problem Set 2
Physics 207: Lecture 3, Pg 32