Ma 116  Calculus II
Hw1
(Solutions)
Due: Feb 9, 2011
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:
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1. An object with mass
m
is dropped from rest at time
t
= 0. A model for the object’s velocity
v
at time
t
, that takes air resistance into account, is given by the differential equation,
dv
dt
+
c
m
v
=

g
Here
g >
0 is the gravitational acceleration and
c >
0 is a positive constant associated with
the air resistance.
•
Confirm that
v
=

mg
c
1

e

ct/m
is a solution to the differential equation and
satisfies
v
(0) = 0.
•
Evaluate lim
t
→∞
v
and give a physical interpretation of this limit.
•
Now fix
t
and evaluate the limit
lim
c
→
0
+
v
. What is the physical interpretation of this
limit?
Solution
:
v
=

mg
c
1

e

ct/m
and
dv
dt
=

ge

ct/m
Substitute into lefthand side of the differential equation:
dv
dt
+
c
m
v
=

ge

ct/m

g
1

e

ct/m
=

g
Solution Confirmed!
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 Fall '11
 Duboski
 Calculus, Radioactive Decay, lim, sterile males

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