Name: Carl W.
I am a parent and was discussing escape velocity with my sons. The
discussion turned to the fact that you would in theory never get out
of a planets gravitational field (assuming it diminishes with distance
but does not vanish). This being the case, if an object was launched
from the surface at a velocity, it would immediately begin to slow down
and given enough time would reach zero velocity and at that point begin
to fall back to the object and would not in fact escape. Is there
something wrong with this argument? or is escape velocity more
complicated than simply looking at one (actually both) objects gravity?
You are absolutely right in saying that the earth's gravitational field
diminishes with distance, but does not vanish, and in saying that an
object launched from the earth with a given velocity will slow down
due to the earth's gravitational field.
However, given a high enough velocity, the object will continue to
increase its distance from the earth indefinitely. The velocity which
will just allow the object to go to infinity (in principle) is called
the escape velocity. It can be calculated by finding the velocity
which gives the object a kinetic energy just equal in magnitude to its
(negative) potential energy at the earth's surface. This makes the
total energy (kinetic plus potential) just equal to zero. Then when
the object is infinitely far away when its potential energy and
kinetic energy are both just about zero and so the total energy is
still zero. Energy is conserved!
The velocity needed to escape is just about 11.2 km/sec or
Best, Dick Plano...
You have the argument exactly right. Escape velocity is the speed that
just barely gets you to infinity. Of course it would take an infinite
time to get there -- not only because infinity is, uh, infinitely far
away, but also because you're supposed to arrive there with zero
Although, you do in fact always feel a little gravity from a planet, the
planet will not necessarily be able to make you turn around. Since you are
always moving further away, gravity gets weaker and weaker. Every second,
the gravity force takes away less velocity. So long each decrease is
smaller than the previous one, it is possible to never bring the rocket to a
halt. One interesting example is 1 - 1/2 - 1/4 - 1/8 - 1/16 - 1/32 - ....
With every term, more is removed from the original value of one. Still,
every term takes away only half of what is left. The value will never
actually equal zero. If the first term were made less than one, the value
would reach zero. If the first term were two, the value would never reach
one. This is similar to how escape velocity works. If the initial velocity
is fast enough, gravity will keep causing the velocity to decrease without
ever stopping the object.
Dr. Ken Mellendorf
Illinois Central College
Hi, Carl !!
In reality, there must be a velocity beyond it
the body does not return to earth. Below this limit
velocity, the body goes up, stops and returns to the
earth. In this case, all the energy that the body has
is kinetic energy, that is : E1 = (1/2)m.v^2
All this energy is converted in potential energy, or :
E2 = m.g.h If you put E1 = E2, than you have :
v = SQRT ( 2.g.h ). Through this expression, you
determine the velocity of a body to reach a distance
"h". In this case, it was employed the force due to
gravity "g", or :
F = m.g.
If you employ the expression for gravitational force
between two masses M and m you have :
F = G (M.m/R^2)
than it is possible
to determine the potential energy of a body :
E2 = GMm/R.
At the infinitum, the velocity must be zero for this
calculation. Let us put
E1 = (1/2) m. v^2
E2 = G Mm/ R
then you get :
v = SQRT ( 2GM/R ).
This is the scape velocity,
which reaches 40.000 km/h.
But - anyway - let us get back to your question, where you
point out that :
" it would immediately begin to slow down
nd given enough time would reach zero velocity and at that point begin
to fall back to the object and would not in fact escape".
Well - in this case - this point is infinity. And that is why you
can consider that the body will NOT return. Infinity is too
far away from here...
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Update: June 2012