Recitation Bow and Arrow (Work and Kinetic Energy)

Recitation
Bow and Arrow (Work and Kinetic Energy)
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Bow and Arrow (Work and Kinetic Energy)
The bow and arrow is one of the oldest weapons of warfare and new
developments in bow technology could tip the balance in battle. Henry V’s
victory against the French at the famous battle of Agincourt (1415), in which
the English were outnumbered four to one, was due in part to their effective
use of the Welsh (or English) longbow, which allowed the English archers to
shoot more rapidly than crossbow archers and release arrows with sufficient
energy to pierce the armor of the French knights 1 . A modern (1966)
development in bow technology is that of the compound bow, shown to the
right, in which cams (eccentric pulleys) are used to increase the energy stored
in the bow but decrease the force required to keep the string fully drawn.
A bow is just a spring with an unusual shape. As the arrow is drawn back on
the bowstring, the body of the bow flexes and stores elastic potential energy.
However, unlike an ideal spring, the force exerted by the bowstring is not
necessarily proportional to the distance that the bowstring is displaced. The
modern compound bow has eccentric pulleys on either end that cause it to act
like a very non-ideal spring. By design, the force required to pull a modern
bow does not increase linearly with distance. You have just bought a
compound bow at a garage sale and you want to figure out its parameters,
such as how much energy you can store in it, how fast it will shoot an arrow,
how hard it is to hold fully drawn, and how far the arrow will travel. After
taking some measurements, you and a friend head off to the archery range.
Objective: By the end of this activity, you should be able to calculate Work
from knowing the force exerted as a function of position and be able to relate
Work to the change in kinetic energy K.
1. You decide that the easiest way to measure the force required to displace
the bowstring is to hang the bow from the ceiling and then pull down on
the bowstring while standing on a scale. Draw a free-body force diagram
for yourself standing on the scale grasping the string of the hanging bow.
Then write an equation employing Newton’s Second Law and then show
how you can use that equation to determine the upward force exerted by
the bowstring (F bow ) from your scale measurements (F N ).
1 See chapter 3 (“Distance Voices”) of Connections (1978) by James Burke for more details and
to see how improvements in agricultural technology ultimately led to the demise of the longbow
as a weapon. For a literary take on this famous battle, delve into Shakespeare’s Henry V.

2. Using your bathroom scale, you have measured your normal force F N as a function of the
displacement of the bowstring. (Your mass in 80 kg.) From your data, determine the force
exerted on the bowstring F bow and make a plot of F bow vs. Stretch Distance. Be sure to
indicate the scale on the Force axis.
Stretch Distance (m) Scale Force F N (N) Bow Force F bow (N)
0.00 784
0.05 748
0.10 711
0.15 675
0.20 636
0.25 618
0.30 613
0.35 636
0.40 670
0.45 686
0.50 691
3. Show on your graph above how you could use the graph to calculate the Work you did on the
bow by pulling back the string as far as you did (but do not do the calculation itself yet).
Explain below in words.

4. Looking at your force graph, you notice something about the restoring force exerted by your
bow that is quite different from a linear spring for large extensions. What is this difference
from a linear spring and why might the bow be designed for this to happen? (Hint: Imagine
yourself holding the bow and trying to aim the arrow as you keep the string stretched.)
5. A friend helping you out looks at your Force versus distance graph and says, “Hey, the force
decreases after about 0.3 meters, so pulling it back any further just reduces the Work you’ve
done.” You look at the graph and say to your friend:
6. Write an analytic expression for the Work done during the interval described by your first
three data points and simplify as much as possible. Call the Force values F 1 , F 2 , and F 3 , and
the uniform spacing between distance measurements d. (Hint: consider the trapezoid shown
below, on the right.)
F
F 2 F 3
F 1
h 1
h 2
x
w
A = ½(h 1 +h 2 )w
Now, if you simplified your expression for the Work for three data points, you should see a
pattern in terms of end points and middle points. How would you modify your expression to
add a fourth data point (F 4 ), also a spacing d apart?

7. Assuming that all of the work done on the bowstring is stored in the bent bow, determine the
amount of energy stored in the bow at full extension (0.5 m). Call this energy, E bow-f .
(Generalize what you did in question 6 above for more than 3 & 4 data points.)
CHECK WITH AN INSTRUCTOR AT THIS POINT
8. You want to estimate the release speed of an arrow (which you have measured to have a
mass of M = 0.02 kg) launched horizontally. You estimate that about 80% of the energy in
the bow goes to kinetic energy K of the arrow. What is the speed of the arrow as it leaves the
bow? First put your answer in terms of E bow-f and M, then solve for the numerical value.
Where did the other 20% of E bow-f go?

9. If you aim the arrow directly horizontal and shoot at a target a distance X = 30 meters away,
how far above or below the bulls-eye (1 meter below the arrow’s release height) will the
arrow hit the target? (Neglect the Drag force from the air on the arrow.) [Solve first using
variables and then calculate the numerical value.]
10. When you reach the target, you measure that the arrow penetrated the target by a distance of
D = 0.05 m. From this, you are able to determine that the average 2 force exerted by the arrow
on the target as it pierced the target was: (First, work it out using variables and then calculate
the numerical value.)
CHECK WITH AN INSTRUCTOR AT THIS POINT
Finally, you notice that while you aimed the arrow horizontally, it wasn’t pointing horizontally
when it hit the target; rather, the arrow seemed to point along its trajectory. You wonder why that
is – that is, what mechanism turned the arrow so that the arrowhead always pointed in the
direction it was moving – and you wonder whether that would also happen if you shot the arrow
in a vacuum. “I’ll have to research that,” you say to your friend as you pull out the arrow.
2 This is the average by distance, not a time average; the average force by distance and the
average force by time are only exactly equal when the average speed is half the impact speed.