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Re: [Phys-l] conservation versus constancy

Let me propose a slightly different example than gravity to get your
thinking going on the slingshot effect. Imagine a charged object with a
small mass moving quickly to the right directly at a similarly charged
object with a much larger mass moving slowly to the right. The center of
mass is moving toward the right also with a speed dominated by the speed of
the very massive charge. The charges both slow down relative to the center
of mass because of the mutual repulsion. They come to some minimum distance
between them and reverse directions relative to the center of mass. They
separate, and when the separation is large both objects have their original
speeds relative to the outside worlds - the light charge is moving quickly
to the right and the heavy charge is moving slowly to the right. There is no
slingshot effect.

Now imagine that as the charges get close to each other that their
convergence is fast enough and that their directions of travel are parallel
but slightly offset from each other - to allowi them to pass by each other.
The mutual repulsion again forces them apart and they eventually regain
their original speeds relative to the center of mass. But this time, the
lighter charge is moving away from the center of mass to the right instead
of to the left. To the outside world, the lighter mass has its original
speed to the right PLUS the continued steady motion of the center of mass
(determined mostly by the heavier mass. So the speed of the heavier charge
has been (almost) added to the speed of the lighter one. If you sit down and
do a sketch of this, you will quickly convince yourself that because of the
motion relative to the CM both incoming and outgoing that in actuality twice
the speed (almost) of the CM (or heavier charge) is added to the lighter
charge. This is the essence of the slingshot effect. It requires a dance
where the objects basically swap their relative positions. The CM just
follows its original motion.

The gravitational case just replaces repulsion with attraction and, to me,
is a little harder to visualize, but the physics is basically the same.

Bob at PC

-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-
bounces@carnot.physics.buffalo.edu] On Behalf Of Robert Cohen
Sent: Tuesday, October 17, 2006 10:04 AM
To: Forum for Physics Educators
Subject: Re: [Phys-l] conservation versus constancy

I'm trying to fit this "sling shot" example within the following
perspective. If we take N2L in its usual form:
Fnet = m dv/dt
And integrate with respect to position to find the average force (with
respect to position), we get
Fnet,avg dot Delta x = Delta (1/2 mv^2)
Defining the right-hand-side as the change in the object's kinetic
energy (macro), I find that in a collision the KE "lost" by one object
is "gained" by the other object only if the total displacement of each
object during the collision is the same (since the average net force on
each object is equal and opposite; assuming no other forces contributing
to the net force).

Is this a valid way of looking at things? If so, during the "sling
shot" is the displacement of each object the same during the
"interaction"?

Also, in this interaction is the relative motion the same before and
after? I thought that if KE is the same before and after than the
relative motions are the same before and after (i.e., v_1,i - v_2,i =
v2,f - v1,f). Is this wrong?

----------------------------------------------------------
Robert A. Cohen, Department of Physics, East Stroudsburg University
570.422.3428 rcohen@po-box.esu.edu http://www.esu.edu/~bbq

The slingshot effect is elastic - i.e., the KE before equals
the KE after.
The mass of the planet is so much more than the mass of the
satellite that one sees a visible change in the speed of the
satellite but not the planet.

It's just like a bowling ball striking a stationary tennis
ball. The bowling continues on with no visible change in
speed, whereas the tennis ball moves at twice the speed of
the bowling ball (a slingshot in one dimension!)
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