MA 36600 LECTURE NOTES: FRIDAY, JANUARY 16
Solutions of Some Differential Equations
Gravity without Air Resistance.
Say that we have a mass
m
at a position
x
=
x
(
t
). We wish to discuss
its motion under the influence of gravity. We assume for simplicity that the object is near sea level, at the
surface of the Earth.
#1: Our variables are time
t
and position
x
. Note that
t
is an independent variable, and
x
=
x
(
t
) is a
dependent variable.
#2: The principle that underlies the problem is Newton’s Law of Gravity i.e.
F
=
−
m g
. (This is only
true for a mass that is near sea level!)
#3: Our initial conditions are
x
(0) =
x
0
and
v
(0) =
v
0
.
The initial value problem is
m
d
2
x
dt
2
=
−
m g
Di
ff
erential
Equation
x
(0) =
x
0
v
(0) =
v
0
Initial
Conditions
We will show that the solution is
x
(
t
) =
x
0
+
v
0
t
−
1
2
g t
2
.
To this end, first we consider the velocity
v
=
x
(
t
). We have the initial value problem
dv
dt
=
−
g
Di
ff
erential
Equation
v
(0) =
v
0
Initial
Condition
The di
ff
erential equation is
dv
dt
=
−
g
=
⇒
v
(
t
) =
−
g t
+
C
for some constant
C
. (You can see this by considering
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 Spring '09
 EdrayGoins
 Differential Equations, Equations, Constant of integration, Boundary value problem

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