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Regarding Ludwik's problem statement:
Here is a problem to help them memorize the situation.
A rifle is fired vertically down from a tall building. The
bullet, initially at rest, leaves the barrel with the velocity
of 300 m/s. The distance traveled inside the barrel is one
meter.
a) How large is the average acceleration inside the barrel?
This question is unanswerable until we know what the form of the
acceleration profile a(x) is inside the barrel. Here a(x) is the
acceleration of the bullet at location x inside the barrel a distance
x from the cartridge (i.e. from the place where it starts to move).
Different forms of the a(x) function will result in different (time)
average acceleration a-bar for same exit speed V and same barrel
length L. For instance, if the acceleration is constant (a very
unrealistic assumption) inside the barrel then we find that
a-bar = (V^2)/(2*L). But if we assume that the a(x) function is
linearly decreasing with increasing x from a maximum value at x = 0
to zero at x = L at the end of the barrel we find that, instead,
a-bar = (2/PI)*(V^2)/L. This form of the profile gives an a-bar
which is 4/PI times larger than for the uniform acceleration profile
case. I expect that in real life even the linear decrease a(x)
profile may be unrealistic. If the profile decays faster than
linearly then the value of a-bar will be even higher than its value
for the case of linear decrease. If the bullet's explosive charge is
too large, or if the barrel is too short then the bullet's
acceleration profile will tend to have nearly a jump discontinuity at
the mouth of the barrel due a side splatter of the gases of the
muzzle flash.
b) How large is the acceleration outside the barrel (ignore
air resistance).
This is a much more straightforward and doable calculation. But a
bullet of a few grams whose speed is comparable to the speed of sound
will likely *not* have a negligible air resistance.
They must know that "free fall" is a situation in which
weight is the only acting force.
Later (when Fnet=m*a is known) you can add the third
question. Giving the bullet's mass (say 10 grams) ask:
c) How large would the acceleration be in water if the
water resistance force were 100 N?
Of course this made up number ignores the real changes in the
dynamics of the explosion that take place because the explosive
charge is propelling not only the bullet but a whole column of water
at least as long as the barrel. Since all this extra mass greatly
slows down the bullet propulsion process it seems that the a(x)
profile will be greatly modified in a way that is significantly
different than by just subtracting a constant background due to a
constant water resistance force contribution. Also, once the
bullet exits the barrel the water resistance force on it is
strongly dependent on the current velocity of the bullet and is
therefore not a constant. This makes the bullet's acceleration
outside of the barrel a complicated function of its position (and/or
the elapsed time).
I certainly sympathize with Ludwik's desire to make the problem
simple enough so it can be done by the intended introductory
students who may be having trouble undertanding the concept of
acceleration itself (let alone the effect of a complicatedly
varying acceleration on the problem). It's just that finding an
illustrative example that it both simple enough and yet still
realistic can be a very tricky proposition.
Ludwik Kowalski
David Bowman
dbowman@georgetowncollege.edu