Message-Id: <l03102802b14ef1212cc7@[209.142.17.104]>
In-Reply-To: <n1320474301.2110a@Tesla.exploratorium.edu>
Date: Mon, 6 Apr 1998 16:29:31 -0700
To: "Pinhole Listserv" <pinhole@exploratorium.edu>
From: Ron Wong <ronwong@inreach.com>
Subject: Re: escape velocity
emile posted the following question:
>I'm still a bit confused about this whole escape velocity thing - if the force
>of gravity always acts upon any two objects at any distance, how can we say
>that they can be going fast enough away from each other to never be pulled
>back together? If only one force is acting on two object, those objects will
>accelerate because of that force, however small, right?
If we're talking just the pair of forces acting between these two objects,
you're right.
But infinities allow a us to draw conclusions that don't seem to make sense
unless viewed from a particular point of view.
For instance, take this issue of escape velocity. If we want to send
something away from us so that it doesn't return and we want to do so by
imparting it with just the right speed at the very beginning of its trip
(the escape velocity) then we have to figure how much work is needed to
give it the kinetic energy it needs to make good its escape after it leaves
us.
In order for it not to return we'll have to send it an infinite distance
away. Since the distance involved is an infinite one , you'd think it would
require an infinite amount of work and thus an infinite amount of energy.
Such is not the case.
The force needed to overcome the gravitational attraction of the body that
the mass is escaping from falls off very rapidly by the inverse square of
the distance and will reach zero when the mass is an infinite distance
away. So, as one gets larger, the other gets smaller. In fact, the force
gets smaller far more rapidly than the distance gets larger. As a result,
when you sum up all the work required (the product of the force and
distance), you end up with a finite value for the work done moving a mass,
m, an infinite distance from another mass, M.
For instance, if M is a sphere and the other mass, m, is R meters from M's
center, then the work done carrying m an infinite distance away is found by
means of the formula, Work = GMm/R. Where G = Universal Gravitation
Constant = 6.67 X 10^-ll Joules-meter/kilogram^2
If you were to do this amount of work on a mass, m, in order to send it on
its way to infinity, than its kinetic energy at the start (0.5mv^2) would
equal the work done on it and
(0.5)mv^2 = GMm/R
Notice that the mass that is being sent away drops out of the equation and
we are left with
v = Square root(2GM/R). No matter what we send away, this must be its speed
if it wants to escape mass M from an initial distance, R, from M's center.
If you plug in G, the mass of the earth (5.98X10^24 kg) and its radius
(6.38X10^6 m) into this formula and you can see where the figure of 11.2
km/sec for the escape velocity of an object on the surface of the earth
came from.
If one of the two objects mentioned in Emile's original question was the
earth and the other was something at rest an infinite distance away, than
the earth's force of gravity would indeed tug on it sending it accelerating
towards the earth. Upon impact, it's speed would be.....11.2 km/sec!
By the way, "escape velocity" is the speed the object must have in order to
escape without any further assistance. If forces can be applied to the
object over a period of time, than it is possible for it to escape without
attaining the initial escape velocity. This concept has been used to send
objects out of our solar system at speeds less than that needed when they
were launched from Earth. Pioneer 10 is the classic example. It was sent
along a path that took it past Jupiter. At that distance the escape
velocity from the sun is less than it would be from our earth's orbit and,
using the gravitational pull of Jupiter, the space probe acquired the
requisite speed needed to escape from our solar system forever.
Speaking of escaping....
chiao
ron