Multiplying both sides of Eq. (1.8) by m/cd and defining a mg / cd yields
m dv
a2 v2
cd dt
Integrate by separation of variables,
cd
a m dt
dv
2
v 2
A table of integrals can be consulted to find that
a
dx 1 x
tan 1
2
x2 a a
Therefore, the integration yields
1 v c
tan 1 d t C
a a m
The initial condition, v(0) = v0 gives
1 v
C tan 1 0
a a
Substituting this result back into the solution yields
1 v c 1 v
tan 1 d t tan 1 0
a a m a a
Multiplying both sides by a and taking the tangent gives
c v
v a tan a d t tan 1 0
m a
or substituting the values for a and simplifying gives
mg gcd cd
v tan t tan 1 v0 (2)
cd m mg
(c) We use Eq. (2) until the velocity reaches zero. Inspection of Eq. (2) indicates that this occurs when the
argument of the tangent is zero. That is, when
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