
Does the amount of stretch of a rubber band
affect the distance a rubber band will travel?
  Answer 1:
Every child who has been in a rubber band battle
instinctively knows the longer you stretch a
rubber band, the farther it will go. But thy is
the case involves some very fundamental ideas of
physics.
Determining how far something can
travel is a classic problem in mechanics.
Physicist would say, when released, the
projectile, i.e. the rubber band, follows a
ballistic trajectory. This is just a way of
saying that when in flight, the rubber band is
only acted upon by gravity and air resistance, and
in this case, the air resistance is almost
negligible.
Thus, after being released, the
flight path is basically determined, and the
control we have are the launch angle and the
initial velocity. The stretch of the rubber band
only affects the initial velocity, so lets explore
that in more detail.
Stretching the rubber
band has the same physical principles as a spring.
As it is stretch, the rubber band applies a force
to return to its equilibrium, or unstretched
point. The amount of force can by calculated by
Hooke's Law,
F=k x,
where k is the spring constant and x is
the distance from equilibrium. The spring
constant is the constant that makes the Hooke's
Law mathematically correct and is usually found by
experimentation. So the more the rubber band is
stretched the more force it applies to return to
equilibrium. So intuitively the farther you pull,
the farther it should go.
To quantify this
intuition, we need to look at the energy.
Stretching the rubber band creates a force;
therefore, it's storing potential energy. The
amount of potential energy it stores is given by
U=1/2 k
x^{2}
. In an ideal
world, all of this potential energy would be
converted into kinetic energy, the energy of a
moving object. Since the kinetic energy is
calculated by
K=1/2 m
v^{2},
where m is the
mass of the rubber band, the initial velocity, v,
is given by
v=[√ (k/m)]
x.
Now I did say in an ideal
world, since some of the potential energy is lost
to heat, but not a significant amount.
So
the longer you stretch the rubber band, the more
initial velocity, meaning the farther it should
fly assuming you keep the angle the same. Well,
there is one last caveat. Rubber bands and
springs have what is known as an elastic limit.
If you pull it past a certain point, the physical
properties of the rubber band changes, decreasing
the spring constant, meaning a smaller initial
velocity and a shorter distance traveled.
  Answer 2:
That sounds like a good question for your
students.If you ask them to 'vote' on the
question, how many will say Yes and how many will
say No? My sons junior high science teacher Mr.
Winn asked a lot of questions like that.
I
did an experiment with a rubber band, and it
landed 9 floor tiles away when I stretched it a
little [9 ft] and it hit the door when I stretched
it more [more than 9 ft; I didn't measure]. I'm
sure your students would be delighted to do
research on the question, and I'm guessing you'd
rather have them do their research at home than in
the classroom.
The grad students and post
docs in the lab discussed the question over coffee
for 20 min or so and got involved with questions
such as terminal velocity, where they decided
that, in some cases, there might be a stretch that
would make the rubber band go as far as possible,
given that air resistance would slow it down; and
more stretch wouldn't make it go farther. I'm not
convinced, because I think if I do more work
stretching the rubber band; it will travel a
larger distance.
The students made an
interesting point: Stretching the rubber band
gives it Potential Energy, like putting a rock on
the top of the hill. The Potential Energy is
converted to Kinetic Energy when you let go of the
rubber band or push the rock so it rolls down the
hill.
Best wishes,
  Answer 3:
The distance traveled DOES depend on the amount
the rubber band is initially stretched. One easy
way to see this is by means of an energy argument:
The potential energy stored in the rubber band is
proportional to the square of the stretched
length. Once released, this energy is converted to
kinetic energy of the rubber band, therefore, the
longer it is stretched, the more kinetic energy it
gets, and thus , the farther it travels. I'm
assuming the holding hand is at rest during the
release of the rubber band, otherwise you would
need to add the mechanical work provided by the
holding hand's displacement during launch. One way
to verify this would be by using a(stationary)
slingshot. The farther the rubber band is
stretched, the farther will the missile travel (as
any 4th grade nonpoliticallycorrect squirrel
hunter might attest).
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