homework 13 – BAUTISTA, ALDO – Due: Feb 22 2006, 4:00 am
1
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Question 1
Part 1 of 3.
10 points.
A block starts at rest and slides down a
frictionless track except for a small rough area
on a horizontal section of the track (as shown
in the figure below).
It leaves the track horizontally, flies through
the air, and subsequently strikes the ground.
The acceleration of gravity is 9
.
81 m
/
s
2
.
μ
=0
.
2
1
.
4 m
441 g
h
2
.
1 m
4
.
78 m
9
.
81 m
/
s
2
v
At what height
h
above the ground is the
block released?
Correct answer:
5
.
10005
m (tolerance
±
1
%).
Explanation:
Let :
x
= 4
.
78 m
,
g
= 9
.
81 m
/
s
2
,
m
= 441 g
,
μ
= 0
.
2
,
‘
= 1
.
4 m
,
h
2
=

2
.
1 m
,
h
=
h
1

h
2
,
and
v
x
=
v .
μ
‘
m
h
h
1
h
2
x
g
v
Basic Concepts:
Conservation of Me
chanical Energy
U
i
=
U
f
+
K
f
+
W .
(1)
since
v
i
= 0 m/s.
K
=
1
2
m v
2
(2)
U
g
=
m g h
(3)
W
=
μ m g ‘ .
(4)
Choosing the point where the block leaves the
track as the origin of the coordinate system,
Δ
x
=
v
x
Δ
t
(5)
h
2
=

1
2
g
Δ
t
2
(6)
since
a
x
i
= 0 m/s
2
and
v
y
i
= 0 m/s.
Solution:
From energy conservation Eqs. 1,
2, 3, and 4, we have
1
2
m v
2
x
=
m g
(
h

h
2
)

μ m g ‘
v
2
x
= 2
g h
1

2
μ g ‘
h
1
=
v
2
x
2
g
+
μ ‘
(7)
h
2
=

1
2
g t
2
(6)
x
=
v
x
t .
(5)
Using Eq. 6 and substituting
t
=
x
v
x
from
Eq. 5, we have
h
2
=

1
2
g
x
v
x
2
,
so
v
2
x
=

g x
2
2
h
2
.
(8)
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homework 13 – BAUTISTA, ALDO – Due: Feb 22 2006, 4:00 am
2
Using Eq. 6 and substituting
v
2
x
from Eq. 8,
we have
h
1
=

g x
2
2
h
2
2
g
+
μ ‘
=

x
2
4
h
2
+
μ ‘
(9)
=

(4
.
78 m)
2
4 (

2
.
1 m)
+ (0
.
2) (1
.
4 m)
= 3
.
00005 m
,
and
h
=
h
1

h
2
= (3
.
00005 m)

(

2
.
1 m)
=
5
.
10005 m
.
Question 2
Part 2 of 3.
10 points.
What is the the speed of the block when it
leaves the track?
Correct answer: 7
.
30529 m
/
s (tolerance
±
1
%).
Explanation:
From Eq. 8, we have
v
x
=
s

g x
2
2
h
2
=
s

(9
.
81 m
/
s
2
) (4
.
78 m)
2
2 (

2
.
1 m)
=
7
.
30529 m
/
s
.
Alternate Solution:
From Eq. 7, we have
v
x
=
p
2
g
(
h
1

μ ‘
)
=
2 (9
.
81 m
/
s
2
)
h
(3
.
00005 m)

(0
.
2) (1
.
4 m)
i
1
/
2
=
7
.
30529 m
/
s
.
Question 3
Part 3 of 3.
10 points.
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 Force, Friction, Mass, Work, Cos, Correct Answer, – BAUTISTA

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