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Physics 221
Fall 2008
Homework #3 Solutions
Chs. 3 and 9:13
Due Tues, Sept 16, 2008
3.1 A particle has a position (
x
,
y
) in the
x

y
plane versus time
t
given by
x
= (0.100 m/s
2
)
t
2
y
= (0.000500 m/s
4
)
t
4
(a) At
t
= 10.0 s, how far is the particle from the origin of the
x

y
coordinate system?
At that time,
what is the angle of the position vector of the particle with respect to the positive
x
axis?
At
t
= 10.0 s, one has
x
= (0.100 m/s
2
)
t
2
=
(0.100 m/s
2
)(10.0 s)
2
= 10.0 m
y
= (0.000500 m/s
4
)
t
4
=
(0.000500 m/s
4
)(10.0 s)
4
=
5.00 m .
The distance of the particle from the origin is
r
=
x
2
+
y
2
=
(10.0 m)
2
+
(5.0 m)
2
=
11.2 m .
The angle of the particle’s position with respect to the positive
x
axis is
±
=
arctan(
y
/
x
)
=
arctan[(5.00 m) / (10.0 m)]
=
26.6°
.
(b) Derive an expression for the path of the particle,
i.e.
, what is
y
(
x
)?
What is the shape of this
path called?
Plot the path,
i.e.
, plot
y
versus
x
for the interval
t
= 0 to
t
= 10.0 s.
Use a
spreadsheet program to make the plot if you can.
From the statement of the problem, solve for
t
2
from the expression for the position of
x
:
t
2
= (10.0 s
2
/m)
x
.
Substitute this into the expression for
y
(
t
) to get
y
=
(0.000500 m/s
4
)[(10.0 s
2
/m)
x
]
2
=
(0.0500/m)
x
2
.
Thus the shape of the path is a parabola.
As noted in part (a), when
t
= 10.0 s, one has
x
= 10.0 m.
A plot of
y
(
x
) follows:
0
1
2
3
4
5
02468
1
0
y (m)
x (m)
Problem 3.1(b)
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(c) Derive expressions for the
x
and
y
components of the velocity of the particle versus time.
What
are the velocities of the particle at times
t
= 0.0 s and
t
= 10.0 s?
Taking the time derivatives of the positions given in the problem yields:
v
x
=
dx
dt
=
d
[(0.100 m/s
2
)
t
2
]
dt
=
(0.200 m/s
2
)
t
.
v
y
=
dy
dt
=
d
[(0.000500 m/s
4
)
t
4
]
dt
=
(0.00200 m/s
4
)
t
3
Thus at
t
= 0 we have
v
x
=
v
y
= 0.
At
t
= 10.0 s we have
v
x
=
v
y
= 2.00 m/s.
(d) Derive expressions for the
x
and
y
components of the acceleration of the particle versus time.
What are the accelerations of the particle at times
t
= 0.0 s and
t
= 10.0 s?
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This note was uploaded on 01/28/2010 for the course PHYSICS 221 taught by Professor Johnson during the Fall '06 term at Iowa State.
 Fall '06
 Johnson
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