Midterm
11
Twovectors a and b are placed on a square grid w a cellside
length of 1 shown in the figure belowYou may assume thatthe unit
vector i points directly tothe right alongthe positive x axis and theunit
vector J points directly up along the positive axis as isconventional
y
0 denotes the
origin.x
a
̅
theformat
Express a and b in componentform Youranswersshould be of
overrightharpoon
a
y
xij
overrightharpoon
2
aij
overrightharpoon
rfloor
iota
b
overrightharpoon
b what is the magnitude of a What is the angle that at
respect to the l direction
overrightharpoon
subtendsw
overrightharpoon
right
sqrt
left
4
5
2
a1
theta
right
frac
left
tan
2
1
Angle
theta
right
frac
circ
left
tan
0
8
2
1
approx
circ
6
0
8
4
3
5
2
1
2
b
a
cj
c Draw the vector overrightharpoon in the diagram aboveandfind the
magnitude of c Alsofind the angle I subtends w respect to the
overrightharpoon
T direction
overrightharpoon
right
sqrt
left
C
9
4
3
2
1
right
left
2
b
a
c
i
j
3
2
cij
overrightharpoon
overrightharpoon
theta
right
frac
circ
left
tan
3
2
7
2 The velocity of a particle on a line in a certain coordinate system is
given by
right
left
m
V
6
8
2
x
s1t
where t is in seconds
a Plotthe velocity vs time cure for the time interval 02 teds In your
plot label and indicate the acceleration and indicate the displacement
st
of the particle from time 0
t to 3
varphi
frac
X
V
1
a
overrightarrow
l
m
V
6
0
4
8
3
2
x
s1
right
left
4
1t
a
6
2
1
a
overrightharpoon
overrightarrow
a
overrightarrow
displacement
8
1
accelerationis
derivative of
velocity
Delta
right
int_
left
6
0
8
3
2
d
x1t
displacementarea undervelocity
curve
b If the particle is located at
nicefrac
m
6
2
d
a
v
st
overrightarrow
right
left
m
9
4
8
3
5
2
7
1
time O
st determine the
position as a function of time Calculate the positionof the particle
at 3
st
position isintegral of
m
0
3
d
xt
right
int_
left
m
X
6
0
8
3
2
d
x1rt
velocity3
Youthrow a ball at angle w respect tothe ground It leavesyour hand
at a height of yi above the ground If the launchspeed is 10msdetermine
ominus
a what is the maximumheight of theball if theta
circ
0
3 and
Given
launchspeed
0
3
Angle theta
Height m
y1i
m
V
0
s1
m
9
8
2
g
s
downarrow
right
left
V
0
2
g
yi
downarrow
right
left
6
9
0
8
5
2
y1
right
left
6
9
8
5
2
y1
approx
right
circ
cos
left
sin
m
V
6
0
8
3
5
y
x
s1
At top
V
0
y
m
y1i
b what.isherangeoftheballif theta
circ
0
3 and
m
y1i
Delta
right
frac
left
V
2
g
y1fti
0
y hits ground
right
frac
left
9
0
8
5
2
1t
9
0
4
5
2
1t
9
0
4
5
2
1t
sqrt
pm
4
2
b
a
c
UseQuadraticFormula
right
sqrt
left
pm
6
9
4
5
2
1
matrix
begin
sqrt
end
pm
6
9
4
8
5
s1t
2a
Range times
X
V
t
right
left
6
9
8
1
m
0
3
1
C Giventhe case where 0
youwish to throwtheball at an angleOmthat
y
i
will result inthe largestpossiblerangeWhat isthis angle Remark onhow
this angle would or wouldn'tchange w varying values of launchspeed
and freefall acceleration Hint
nicefrac
cos
sin
2
d
x
theta
frac
sin
V
0
R
2
g
sin
m
9
0
4
5
2
1
theta
circ
1 Susan leaves Pittsburgh at 9AMtraveling west at a constant 40mphBob
leaves Chicago at 10AM driving east at a constant 60mpitWhere do
they meet forlunch given Chicago is 400miles westof Pittsburgh
Susan
leaves9AM
speed 40MPHWest
east400
Bob
leaves 10AM
speed 60MPH
Vrelative
m
H
6
0
4
p
1
times
0
4
d
s1
6
4
0
3
miles
milesremaining when
Bobleaves
t 360mmiles
0
frac
0
1p1t
6
3
h
u
o
sr
6
0
3
60min cdot
ominsSusan
right
left
6
0
4
8
1
Pittsburgh2
2kg
You
travels 1 hour more
6
3
h ours
6
4
h
u
o
s
1r
4miles west of
You pulla box w mass m
respect tothe horizontal
on a slope that makesangle theta
0
3 w
pull alongthe tangent totheslope
F
neg
pull
overrightharpoon
m
theta
lceil
a the coefficient ofstaticfriction between the box and theslope Ns 0.2
Whatis the minimum pullingforce you mustapply tothe box tomove it
Fpull
theta
cos
sin
m
N
g
s
overrightharpoon
m
9
8
2
g
k
s
right
circ
cos
left
sin
N
6
9
0
8
3
5
2
1
theta
N
0
3
2
s
b Pulling w the force you determined in parta the box slipsand moves
up the slope The coefficient ofkineticfriction between the boxand the
slope is N
K
0
1 How far up the slope have you pushed the box 2 seconds
after it slips
Forcesdown the slope
9
8
g
kineticfriction
theta
right
cos
left
m
N
9
0
8
3
2
g
7
k1f
N
9
8
5
7
1
NetForce
Acceleration
Fnet Fpull 1 downwardforces
aFnet
m
frac
m
0
8
5
2
7
s1
Distance after25
N
3
5
2
7
1
right
frac
left
m
0
4
8
5
2
7
1
4
5 over a
3 A projectile w mass m
is launched at an angletheta
g
k
1
basin that is Delta
x 10m wide You are intially at a height of y
1 20mabove
the basin at the very edge ofthe ledgeand you aretrying to land the
projectile on a ledge to the right ofthe basin that is
abovethe basin
m
0
2
y
1
In this problem use a freefall accelerationmagnitude of m
9
8
2
g
s andassume
the positive x axis points totheright
V
0
theta
1
theta
4
5
m
0
2
y1
m
0
2
y1
Delta
m
0
x
1
sqrt
frac
X
V
0
2
o
x1t
cdot
sqrt
frac
V
6
0
2
c1t
rightarrow
sqrt
frac
m
V
o
0
g
1
2
yxt
sqrt
frac
V
9
0
4
2t
substitute
right
sqrt
frac
left
V
9
0
4
2
1
right
frac
left
V
9
0
4
2
1
frac
O
V
9
0
8
3
2
frac
m
V
6
0
3
5
2
7
s
V
9
0
8
3
2