Problem
Set 1: Problem 5.
Problem:
The acceleration of a particle is
a
=
−
λ
√
v
,where
λ
is a constant. At time
t
=0
the particle
is located at
x
=0
and its velocity is
v
=
V
. Also, when its position is
x
=
f
the particle’s velocity is
v
=
1
4
V
.
(a) Determine the constant
λ
.
(b) At what time
τ
does the particle come to rest?
Solution:
Since we know the acceleration as a function of velocity, the obvious first step in solving this
problem is to begin with the differential equation relating acceleration and velocity, viz.,
a
=
v
dv
dx
=
⇒
adx
=
vdv
(a)
For the given acceleration, we have
−
λ
√
vdx
=
vdv
=
⇒
dx
=
−
1
λ
√
vdv
Then, integrating, we have
8
x
o
dx
=
−
1
λ
8
v
V
√
vdv
=
⇒
x
=
−
2
3
λ
v
3
/
2
e
e
e
e
v
=
v
v
=
V
Thus, the particle’s position as a function of its velocity is
x
=
−
2
3
λ
p
v
3
/
2
−
V
3
/
2
Q
We can now determine the value of
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This note was uploaded on 05/12/2010 for the course AME 301 taught by Professor Shiflett during the Spring '06 term at USC.
 Spring '06
 Shiflett

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