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Physics 2211001, Fall 2008
Gravity
Due at 11:00pm on Saturday, October 4, 2008
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Escape Velocity
Learning Goal:
To introduce you to the concept of escape velocity for a rocket.
The escape velocity is defined to be the minimum speed with which an object of mass
must move to escape from
the gravitational attraction of a much larger body, such as a planet of total mass
. The escape velocity is a function
of the distance of the object from the center of the planet
, but unless otherwise specified this distance is taken to be
the radius of the planet because it addresses the question "How fast does my rocket have to go to escape from the
surface of the planet?"
Part A
The key to making a concise mathematical definition of escape velocity is to consider the energy. If an object is
launched at its escape velocity, what is the total mechanical energy
of the object at a very large (i.e., infinite)
distance from the planet? Follow the usual convention and take the gravitational potential energy to be zero at very
large distances.
Hint A.1 Consider various cases
Hint not displayed
ANSWER:
=
0
Consider the motion of an object between a point close to the planet and a point very very far from the planet.
Indicate whether the following statements are true or false.
Part B
Angular momentum about the center of the planet is conserved.
ANSWER:
true
false
Part C
Total mechanical energy is conserved.
ANSWER:
true
false
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Part D
Kinetic energy is conserved.
ANSWER:
true
false
Part E
The angular momentum about the center of the planet and the total mechanical energy will be conserved regardless of
whether the object moves from small
to large
(like a rocket being launched) or from large
to small
(like a
comet approaching the earth).
ANSWER:
true
false
What if the object is not moving directly away from or toward the planet but instead is moving at an angle
from the normal? In this case, it will have a tangential velocity
and angular momentum
. Since angular momentum is conserved,
for any
, so
will go to 0 as
goes to infinity.
This means that angular momentum can be conserved without adding any kinetic energy at
. The
important aspect for determining the escape velocity will therefore be the conservation of total mechanical
energy.
Part F
Find the escape velocity
for an object of mass
that is initially at a distance
from the center of a planet of
mass
. Assume that
, the radius of the planet, and ignore air resistance.
Part F.1 Determine the gravitational potential energy
Part not displayed
Part F.2
Determine the kinetic energy
Part not displayed
Hint F.3
Putting it all together
Hint not displayed
Express the escape velocity in terms of
,
,
, and
, the universal gravitational constant.
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 Fall '08
 DUNCAN
 Gravity, Mass, Gravitational forces, Celestial mechanics

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