Version 048 – Final Exam – chelikowsky – (59005)
1
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001
10.0 points
Given:
The battleship and enemy ships
A
and
B
lie along a straight line.
Neglect air
friction.
A battleship simultaneously fires two shells
(with the same muzzle velocity) at these two
enemy ships.
battleship
A
B
If the shells follow the parabolic trajectories
shown in the figure, which ship gets hit first?
1.
need more information
2.
A
3.
B
correct
4.
both at the same time
Explanation:
The time interval for the entire projectile
motion is given by
t
trip
=
t
rise
+
t
fall
= 2
t
rise
,
where
t
rise
is the rising time from 0 to the
maximum height, and
t
fall
the falling time
from
h
to 0. In the absence of air resistance
t
rise
=
t
fall
,
h
=
1
2
g t
2
fall
, or
t
trip
= 2
radicalBigg
2
h
g
.
So the smaller is
h
, the smaller is
t
trip
. In
other words, enemy ship
B
will get hit first.
002
10.0 points
When a metal ball of unknown mass
M
is
suspended from a spring of unknown force
constant
k
, the spring’s equilibrium length
increases by Δ
L
e
. And when the ball is out of
equilibrium, it oscillates up and down with a
period
T
.
Find Δ
L
e
.
The acceleration of gravity is
9
.
8 m
/
s
2
and the period is 0
.
412 s.
1. 12.303
2. 6.76406
3. 9.54223
4. 1.704
5. 12.1289
6. 4.21367
7. 8.06522
8. 4.52608
9. 15.258
10. 13.7039
Correct answer: 4
.
21367 cm.
Explanation:
Let :
g
= 9
.
8 m
/
s
2
,
and
T
= 0
.
412 s
.
Let
L
be the length of the free spring with
out the ball. When a ball is suspended in equi
librium from the spring, the spring lengthens
by Δ
L
e
to generate the tension force
F
S
e
=
k
Δ
L
e
=
Mg,
so
(1)
Δ
L
e
=
Mg
k
.
(2)
Now consider the oscillating ball.
When
the ball is at height
y
above the equilibrium
point, the spring is stretched by
Δ
L
(
y
) = Δ
L
e
−
y
(3)
and has tension
F
S
(
y
) =
k
Δ
L
(
y
)
=
k
(Δ
L
e
−
y
) =
Mg
−
k y.
(4)
Consequently, the net vertical force on the
ball is
F
net
y
=
F
S
−
Mg
=
−
k y,
(5)
which makes the ball oscillate with angular
frequency
ω
=
radicalbigg
k
M
(6)
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Version 048 – Final Exam – chelikowsky – (59005)
2
and period
T
=
2
π
ω
= 2
π
radicalbigg
M
k
.
(7)
We do not know the ball’s mass
M
nor
the spring’s force constant
k
, but given the
oscillation period
T
we may find the
ratio
M
k
=
parenleftbigg
T
2
π
parenrightbigg
2
,
so
(8)
Δ
L
e
=
Mg
k
=
gT
2
4
π
2
=
(9
.
8 m
/
s
2
)(0
.
412 s)
2
4
π
2
=
4
.
21367 cm
.
(9)
003
10.0 points
Given:
Two vectors
vector
A
=
A
x
ˆ
ı
+
A
y
ˆ
and
vector
B
=
B
x
ˆ
ı
+
B
y
ˆ
,
where
A
x
=
−
2,
A
y
= 9,
B
x
= 2, and
B
y
=
10.
Find the
z
component of
vector
A
×
vector
B
.
1.
86
2.
−
2
3.
−
38
correct
4.
−
94
Explanation:
It follows from the definition of cross prod
uct that
(
vector
A
×
vector
B
)
z
=
A
x
B
y
−
A
y
B
x
= (
−
2) (10)
−
(9) (2)
=
−
38
.
004
10.0 points
The distance between the sun and the Earth
is about 1
.
4991
×
10
11
m.
Express this distance in kilometers.
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 Fall '08
 Turner
 Acceleration, Force, Angular velocity, Velocity, mg Newton

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