LAB #2
Escape Velocity
Goal
: Determine the initial velocity an object is shot upward from the surface of the
earth so as to never return; illustrate scaling variables to simplify differential equations.
Required tools
:
dfield
and
pplane
; second order differential equations; convert
ing a second order differential equation with missing independent variable to a first
order differential equation; converting a second order differential equation to a system
of differential equations.
Discussion
Newton’s Law of Gravitational Attraction tells us that in general an object of mass
m
which is
x
miles above the surface of the earth will feel a force of approximately
F
=

0
.
0061
m
1 +
x
4000
2
pounds with the number “4000” an approximation to the radius of the earth in miles
and “0.0061” the acceleration due to gravity measured in miles/sec
2
.
On the other hand, according to Newton’s 2
nd
Law
F
=
m
d
2
x
dt
2
.
Thus equating these expressions for
F
leads to the differential equation
d
2
x
dt
2
=

0
.
0061
1 +
x
4000
2
.
To study this equation using the Student Edition of
Matlab
, we need to change the
units to avoid overly large intervals. Rather than measuring distance in miles, we will
use multiples of the radius of the earth. This amounts to making the substitution
y
=
x
4000
(scaling the dependent variable) resulting in the equation
d
2
y
dt
2
=

0
.
0061
4000
1
(1 +
y
)
2
.
(
A
)
We can further simplify this equation by changing the units of time (scaling the
independent variable). If we let
s
=
0
.
0061
4000
t
1
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then by the Chain Rule
dy
dt
=
dy
ds
ds
dt
=
0
.
0061
4000
dy
ds
and hence
d
2
y
dt
2
=
d
dt
dy
dt
=
d
dt
0
.
0061
4000
dy
ds
d
2
y
dt
2
=
0
.
0061
4000
d
dt
dy
ds
d
2
y
dt
2
=
0
.
0061
4000
d
ds
dy
ds
ds
dt
d
2
y
dt
2
=
0
.
0061
4000
d
2
y
ds
2
Using the last expression, the differential equation
(A)
becomes
d
2
y
ds
2
=

1
(1 +
y
)
2
.
(
B
)
Hence
y
is now viewed as a function of
s
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 Spring '08
 CHO
 Equations, Velocity, Expression, order differential equation

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