Transcript (the terminal velocity is smaller for larger cross

Physics 218: Mechanics
Instructor: Dr. Tatiana Erukhimova
Lectures 6,7
A ball is thrown vertically upward with a velocity of
magnitude v1 from a window at height H. What is the
ball’s position and velocity at any time moment? How
long does it take to reach the highest point? How long
does it take to reach the ground? What is the velocity of
the ball when it hits the ground?
A ball is thrown vertically upward with a velocity of
magnitude v1 from a window at height H. In addition to
gravity acting on the ball there is another force so that
the acceleration in the up direction is –g+t where  is a
constant and t is the time. What is the ball’s position
when the acceleration is zero?
Falling with air resistance
dv
a
 g  kv
dt
Falling with air resistance
dv
2
a=
= g - kv
dt
Terminal Velocity with Coffee Filters
mg - Fr = ma
where Fr is the resistance force.
Fr
a=gm
1. A penny and a quarter dropped from a ladder land at the
same time (air resistance is negligible).
2. A coin dropped in a coffee filter from a ladder lands later
than a coin without coffee filter (the terminal velocity is
smaller for larger cross-section area).
3. A quarter dropped in a coffee filter will land faster than a
penny in a coffee filter (the terminal velocity is larger for
larger mass)
4. Two identical coins dropped in coffee filters of different
diameters land at different times (the terminal velocity is
smaller for larger cross-section area).
Resistance force: Fr = gAv
2
A – area of the projectile
For a spherical projectile in air at g = 0.25 N ´ s /m
STP:
2
4
Terminal velocity:
Fr
a=g=0
m
Fr = mg
gAv = mg
mg
vT =
gA
2
A 70-kg man with a parachute: vT ~ 5 m/s
A 70-kg man without a parachute: vT ~ 70
m/s
Adding Vectors by Components
1. Draw a diagram
2. Choose x and y axes. Choose them in a way that make
your work easier. (E.g. choose one axis along the
direction of one of the vectors so that the vector will
have only one component).
3. Resolve each vector in x and y components
4. Calculate each component using sine and cosine. Be
careful with signs: any component that points along the
negative x or y axis gets a negative sign.
5. Add the x components together to get the x component
of the resultant. Similar for y:
Vx=V1x+V2x+…
Vy=V1y+V2y+…
6. If you want to know the magnitude and direction of
the resultant vector,
V  V V
2
x
2
y
tan  
Vy
Vx
CRAYFISH, SWAN, AND PIKE
(Translation of I. Krylov's fable)
Let crayfish, swan and pike
Draw heavy loaded cart,
Each being just a part
Of harness they dislike.
They try a lot, and everyone
Starts pulling it with zeal;
The problem is that each of them
With his path wants to deal!
The swan makes upward for a cloud,
The crayfish falls behind;
The pike dives sharply in the deep,
And cart moves not from site.
The moral of the verse is that
Accordance should prevail
Amid the people who have plans
To work but not in vain.
Russian fable: Swan, Crawfish, and Pike
Fs
Lake

Despite
their
huge effort the
box does not
move!

River
Fp
Fc
Find Fs and Fc if Fp, θ, and  are
given
 

a) Express the vectors F1, F2, and F in terms of their components.
3
y

F2
2
3

F3
x
1

F1

F4 ,
b) Find the components of the fourth force,
that should be
added for the object to be in static equilibrium.
A cannon at the origin points up at an angle θ with
the x axis. A shell is fired which leaves the barrel
with a velocity of magnitude Vm.
a) When does the shell reach its maximum height?
b)What is the maximum height?
c) What is the range (horizontal distance)?
d)What is the velocity of the shell when it hits the
ground?
A can drops from a magnet just when a bullet is shot from a
gun: Find the angle that the gun must be aimed at to hit the
can.
y
vi
H
θ
L
x
A faulty model rocket moves in the xy-plane (the positive ydirection is vertically upward). The rocket’s acceleration has
components ax(t)=t2 and ay(t)=-t, where =2.50 m/s4, =9.00
m/s2, and =1.40 m/s3. At
 t=0 the rocket is at the origin and has

velocity v0  v0 x i  v0 y j with v0 x  1.00 m / s and v0 y  7.00 m / s.
a) Calculate the velocity and position vectors as functions of time.
b) What is the maximum height reached by the rocket?
c) What is the horizontal displacement of the rocket when it
returns to y=0?
Have a great day!
Chapters 4,5 videos
Reading: Chapter 5
Chapters 3-4 problems
exercises
and