Momentum and Energy Confusion
Date: Winter 2012-2013
The uses in the archery industry of the words 'momentum' and 'kinetic energy' are used interchangeably. A fellow archer and I have had heated debates on the benefits of a slower moving heavier arrow compared to a faster moving lighter arrow. We are hung up on which measurement is more important for a humane kill. A 350 grain arrow moving 330 feet per second has the same momentum as a 330 grain arrow moving 350 feet per second but different kinetic energy. Please help.
I do not know a lot about archery but the Physics applies to any moving body. Momentum is mass times velocity; that's it. So in the example you cite, 350 x 330 = 330 x 350. However, Kinetic energy is 1/2 mv^2. The velocity squared makes a BIG difference. I will calculate the energy of each arrow in metric units, it's just easier for me. I used these conversions:
350 grains = .02267 kg
330 grains = .02138 kg
330 ft/sec = 100.6 m/sec
350 ft/sec = 106.7 m/sec
arrow one = 1/2 (.02267)(100.6)2 arrow one 115 Joules
arrow two = 1/2(.02138)(106.7)2 arrow two = 122 Joules
Having the speed squared makes a big difference. If you double the speed you quadruple the energy. I do not know what constitutes a humane kill as I do not hunt, but I hope this sheds some light on your discussion.
Momentum relates force to time. Kinetic energy relates force to distance. Consider two objects experiencing the same stopping force. An object with twice the momentum requires twice the time to come to a stop. An object with twice the kinetic energy requires twice the distance to come to a stop. A faster but lighter object will slow at a greater rate (larger acceleration), but will have more velocity to ?get rid of?. This results in the same time to stop. Because the lighter object is initially moving faster, it travels further in the first moments. This results in a greater stopping distance. If you want the arrows to take the same amount of time to stop, balance the momentum (mv). If you want the same distance, balance the kinetic energy ( (1/2)mv^2 ).
Dr. Ken Mellendorf
Illinois Central College
Thanks for the question. Momentum and kinetic energy are NOT the same. They are given by different equations and have different units. Momentum is given by P = M*V, where P is the momentum, M is the mass, and V is the speed. Kinetic energy is given by K = 0.5*M*V*V where K is the kinetic energy. Notice that in the equation for K, the speed is squared (V*V) instead of just V in the momentum equation.
You can use the equation above for kinetic energy to calculate which arrow has more kinetic energy. All else being equal, the higher kinetic energy, the more damage can be done to tissue. (Please add emphasis to the "all else being equal" since the factors are not usually equal!) For further reading on the subject, I would look up ballistic gelatin.
I hope this helps. Please let me know if you have more questions.
Realistically, your aim is much more important than small differences
in arrow weight when you are talking about a humane kill. If you hit in
the right spot, the kill will be much more humane, if you miss your
target, you have problems. Aim depends on the shooter and the bow much
more than a few grains difference in the arrow. However, that said, a
lighter arrow will have higher velocity, which means more accuracy
(less drop and less time to hit your target). Obviously there are
extremes where this general way of thinking would no longer hold (e.g.
too light of an arrow can damage your bow), but as long as you're
within recommended guidelines, lighter is going to give you a more
accurate shot, and therefore, more humane kills.
Now, if you want to assume aim is not a factor, that the arrow hits
the exact right spot, and that both arrows hit the target with the
same energy, then the difference in arrows does not matter a lot
(unless you want to get *very* technical). If the arrow is moving
quickly (fast enough that it readily penetrates into the target), then
the penetration depth is the same between your two hypothetical arrows
(penetration depth is proportional to the length off the arrow times
the ratio of the density of the arrow and the animal). If that is not
good enough, and you want to get very technical, then I suggest you
look up the Alekseevski-Tate equations, but that is well beyond the
scope of this web site.
Hope this helps,
It is true that the momentum would be equal, p=mv, and as you correctly
pointed out the energies would be different. E=(1/2)mv^2. This means
the momentum is linearly proportional to the velocity but kinetic
energy is quadratically proportional. This means that for a small
change in velocity, there is a large change in Energy.
Another factor to take in consideration concerning an arrow hitting a
deer is the applied force of the object. The force in this case would
be equal since F=ma where the acceleration is the change in velocity
which would be the final velocity. This is assuming the arrows stopped
moving at the same time after hitting the deer.
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Update: November 2011