arellano (aa39398) – Homework 10 – weathers – (17101)
1
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001
10.0 points
A timevarying net force acting on a 2
.
1 kg
particle causes the object to have a displace
ment given by
x
=
a
+
b t
+
d t
2
+
e t
3
,
where
a
= 2
.
5 m
,
b
= 2
.
3 m
/
s
,
d
=

2
.
4 m
/
s
2
,
and
e
= 0
.
92 m
/
s
3
,
with
x
in meters and
t
in seconds.
Find the work done on the particle in the
first 2
.
6 s of motion.
Correct answer: 69
.
9087 J.
Explanation:
Since the force is time dependent
W
≡
integraldisplay
x
f
x
i
vector
F
·
dvectorx
=
integraldisplay
x
f
x
i
m a dx
=
m
integraldisplay
x
f
x
i
d v
dt
dx
=
m
integraldisplay
x
f
x
i
d v
dx
d x
dt
dx
=
m
integraldisplay
v
f
v
i
v dv
=
1
2
m v
2
f

1
2
m v
2
i
.
Therefore work done on the particle is the
change in kinetic energy. For this case,
W
= Δ
K
=
K
f

K
i
=
1
2
m
(
v
2
f

v
2
i
)
,
where the velocity is found by differentiating
the displacement:
v
=
d x
dt
=
b
+ 2
d t
+ 3
e t
2
v
i
= 2
.
3 m
/
s
,
and
(1)
v
f
= (2
.
3 m
/
s) + 2 (

2
.
4 m
/
s
2
) (2
.
6 s)
+3 (0
.
92 m
/
s
3
) (2
.
6 s)
2
= 8
.
4776 m
/
s
,
(2)
where
t
i
= 0 s and
t
f
= 2
.
6 s
.
Evaluation of the velocity at the initial and
final times gives the desired result.
W
=
K
f

K
i
=
1
2
m
(
v
2
f

v
2
i
)
=
1
2
(2
.
1 kg)
bracketleftBig
(8
.
4776 m
/
s)
2

(2
.
3 m
/
s)
2
bracketrightBig
= (75
.
4632 J)

(5
.
5545 J)
=
69
.
9087 J
.
002 (part 1 of 3) 10.0 points
A 1560 kg car accelerates uniformly from rest
to a speed of 13
.
4 m
/
s in 4
.
93 s.
Find the work done by the net force on the
car during this time interval
Correct answer: 1
.
40057
×
10
5
J.
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 Fall '10
 Weathers
 Energy, Force, Work, Correct Answer, m/s, Arellano

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