Re: [Phys-l] Percent KE retained in perfectly inelastic collision

["Krishna Chowdary" <chowdark@evergreen.edu>]

On Dec 16, 2007 8:16 PM, Michael Edmiston <edmiston@bluffton.edu> wrote:

> The problem is a one-dimensional perfectly-inelastic "dead-on" collision.  Mass one (m1) comes in with initial velocity v1i and strikes stationary mass two (m2).  The two masses stick together and the combined mass (m1+m2) goes off with final velocity vf.  The data for the problem are m1, m2, and v1i and a statement that the collision is perfectly inelastic.  Part-A of the problem asks them to solve for vf.  Part-B of the problem asks them to solve for the percentage of KE retained.
>
> I assumed students would get the percentage by KEf/KEi*100%.  However, one student got the correct answer by vf/v1i*100%.  I first marked it wrong with the comment that the answer was accidentally correct.  But then I did a little algebra and discovered this method is correct in general.  I doubt the student knew that.  I've never seen it before.  Has anybody on the list seen this?  Does this appear in any textbook you are aware of?
.

While I hadn't seen the vf/v1i version before, I've seen and taught
something very similar:  in the type of collision you describe (a
sticking collision between two objects where one object is stationary
before the collision; stating one-dimensional seems redundant to me),
the ratio of the final kinetic energy KEf to the initial kinetic
energy KEi is equal to the ratio of the mass of the initially moving
object to the mass of the combined mass moving after the collision:

KEf/KEi = m1/(m1 + m2).  This is one line of algebra using KE =
p^2/(2m) (I encourage students to decide which version, p^2/2m or 1/2
mv^2 is best to use for a particular problem based on what output they
are aiming for).

Since it's clear from the first part of the problem that m1*v1i = (m1
+ m2)*vf, it's straightforward to see that vf/v1i = m1/(m1 + m2).

It took me much longer to type this than to see this.  I see little
reason why a bright student couldn't have done this on the fly,
noticed this result while doing similar problems previously, or had it
pointed out to them.  In some sense, this could be considered
preferable on an exam, since this little bit of algebra quickly shows
you that you don't have to calculate the KE's, but instead can just
use quantities which are already given in the problem or that you have
already solved for, so save time in calculating something you don't
need.  And if the first part was solved algebraically, all the pieces
seem to be there.

If you asked for (and evaluate based on) showing all work, then the
student just using vf/vi seems incomplete.  How about this:  if the
student had used m1/(m1 + m2) w/o explanation, would that bother you?
If not, then I suggest that vf/v1i requires no further justification,
given supporting algebraic work in the first part of the problem.

I don't care for it, though, as it is only true for this
super-specialized case; it wouldn't work if both objects were moving
before the collision (obviously - what would students use for v1i?).
But it is true for this collision between two objects (and you could
always transform into a reference frame where one of the objects is
stationary, so it actually does become a kind of generally useful
result), so I guess it depends on what rules you have set forth for
students regarding special cases.  I don't have textbooks at home with
me, but I wouldn't be surprised if this is in many textbooks - the
typical introductory book is so full of special case formulas that it
drives me batty.  I don't see why it needs to be in a textbook,
though; if students have worked through a couple of this type of
problem algebraically and paid attention, this result is quickly
noticeable.  If they were drilled on this kind of very common problem
in a previous physics course, it's not at all unlikely that their
previous instructor pointed this out to them or that they saw it
themselves.

Good luck grading your exams!

regards
-Krishna