One of our archer competitors in flight shooting recently made the statement that shooting an arrow much past it’s terminal velocity speed is counter productive. This individual is also a physicist so he is usually pretty accurate with his statement.
For the first time this gives us something to work off of that is not based primarily on trial and error. I have a few questions about terminal velocity.
Everything I read says that objects accelerate at the rate of 9.8 meters per second. Does this hold true all the way up to terminal velocity or does the acceleration slow as it approaches max speed? Would a streamlined dense arrow accelerate at the same rate as basketball would accelerate? What co-efficient of drag is the 9.8 meters per second based on?
Could I drop an arrow 100 ft and get a good idea as to what it’s terminal velocity might be based on a short drop?
Objects accelerate at a constant acceleration only as long as any resistance to moving is negligible. When it is getting close to terminal velocity, the acceleration will be small.
The arrow and basketball, and a feather for that matter, will all do that. They’ll all accelerate at a constant acceleration at speeds that are low enough that they feel no resistance. However, the speed at which resistance starts to be noticeable will be much higher for the arrow, lower for the feather, and in between for the basketball.
I doubt an arrow would get near its terminal velocity in 100 feet, but I think it’d actually be fairly difficult to figure it out theoretically unless one knew some kind of aerodynamic resistance profile for the arrow.
The object only accelerates at 9.8 meters per second^2 when it’s starting to fall (or in a vacuum). The acceleration will decrease to zero as terminal velocity is approached. At the point of terminal velocity, the upward force of drag equals the downward force of arrow mass*9.8 meters per second^2.
What I meant was suppose an arrow reaches 75% of it’s terminal velocity will the last 25% be at the same rate as the first 75%? I am pretty sure it would slow but I just want to make sure. I don’t think the arrows are going high enough to actually reach terminal velocity on the way down.
In ideal spherical-cow land, no object ever quite reaches its terminal velocity, but merely asymptotically approaches it. So yes, it’ll take longer to get from 75% of T.V. to 100% of it than it did to get from 0 to 75%.
I don’t know why it would be counter productive for an arrow to exceed its terminal velocity. That velocity (in air at sea level) is purely derived from the conditions, it is not something inherent to the arrow design. It is dependent on the pull of gravity and the air density. If you were up on a mountain it would have a higher terminal velocity. If the gravity were stronger the velocity would be higher. And what does counter productive mean in this case? Less accurate? The greater draw weight produces a lesser proportional velocity?
A bullet only has a free fall terminal velocity of a few hundred feet per second but it sure is productive fired at a few thousand feet per second.
What your friend may have meant is that in an arcing flight where all you are considering is impact velocity, it doesn’t help to shoot the arrow much higher than the height it needs to attain terminal velocity.
In other words, if you could shoot an arrow at say 60 degrees and hit your target with the arrow going terminal velocity, it would not help at all to use a more powerful bow to shoot the arrow higher while aiming at the same target (say, shooting it twice as fast but at 80 degrees so it goes higher). In those two cases, the arrow will hit at the same speed in the same place, so all a more powerful bow would do for you is increase the hang time of the arrow. It would go higher, then achieve terminal velocity a higher distance from the ground, then hit the ground at terminal velocity anyway.
So in that sense, if you are aiming at a fixed target using a ballistic trajectory like a longbow archer, the most efficient bow would be one that just barely has enough power to ensure that the arrow achieves terminal velocity before impact. Anything more is just a waste of energy that will be converted into heat through atmospheric friction, but will have no greater impact on the target.
However, the heavier bow will cut down flight time because you can use a shallower angle, and it will give you more range if you need it. And if you are shooting in a flat trajectory, the power of the bow will definitely increase the impact speed of the arrow.
There is a very slight chance he may have been confusing terminal velocity with escape velocity. If you can fire an arrow at greater than escape velocity, it won’t start to drop until the speed drops below that point, which will cause it to hit the target well above your planned point.
With an arrow we often sacrifice terminal velocity speed by using a lighter arrow favoring more launch speed. Our physicist feels that the extra speed is quickly dissipated and not worth the sacrifice. There is a point where we are optimum and he seems to feel it is just a bit above terminal velocity. I have yet to find anyone that has been able to accurately factor out the drag to give us a fairly close terminal velocity for a given arrow, they are all very different. That is why I was wondering if I could extrapolate out from a shorter drop of maybe 1 or 2 hundred ft. I doubt the primitive arrows I am shooting go much over 600 ft in altitude.
We are not aiming at fixed target, we are shooting for maximum distance. Between 38 and 45 degrees seems to consistently give best results depending on shooting conditions.
The Earth’s gravity accelerates objects towards it at 9.8 m/s. Anything in the way, like air, will provide a negative acceleration. The maths are covered here and here.
If the arrow is slowing to terminal velocity before it hits, then using a lighter arrow is going to make things worse, because it will have a lower terminal velocity. It will also have less inertia and therefore less penetration power when it hits something,
If you are talking terminal velocity when the arrow leaves the bow, it doesn’t make sense. Terminal velocity is about the steady-state condition after the projectile has expended all its excess energy and is now dominated by the balance between friction and gravity. If the arrow will always be going at terminal velocity when it hits, you would be better off with a heavier arrow shot out of a more powerful bow.
If this isn’t what you are talking about, and you are thinking more along the lines of a faster/lighter arrow slowing down faster and all that, really the best way to figure out what is beat for your situation is to experiment. Take out some arrows of different weight snd shoot them at the target and measure their penetration. You can’t really calculate this because there are about a million factors at play. For example, is the lighter arrow less stiff? Does it need larger fletches to keep it stable? Are there aerodynamic effects like boundary layer tripping coming i to play?
Take out a bunch of arrows, shoot them at targets at different ranges, and measure the results.
This is exactly how we do it. I have made hundreds of arrows over the years and because of small idiosyncrasies of individual arrows it is really hard to find out anything really conclusive. For example I can shoot a 100 grain arrow at 300 fps and a 200 grain arrow at about 240 fps and a 300 grain arrow at maybe 215 fps. Shooting an illegal but very dense carbon arrow the 100 grain will by far go the furthest. But with wood we have to balance out stiffness, weight and shaft diameter. I usually seem to do the best at around 230 grains but a different wood can change that.
Just because we’ve had the same mistake show up twice: The Earth’s surface gravity is not 9.8 m/s (which would be a speed). It’s 9.8 m/s^2 (an acceleration). In other words, an object (with no other forces on it) would change its velocity by 9.8 m/s every second.
I have a question, not related to the OPs post, but I am sure the OP can explain.
What does “100 grain arrow” mean? I assumed it was the weight of the arrow, but 100 grain is less than ¼ oz. Is that correct? Or is some other grain being used?
While the over the entire flight the average lift produced by the fetching becomes negligibly small a drop test wouldn’t accurately capture those drag components or offer the same profile to the air for calculating the maximum distance.
Unless you had enough vertical height for the arrow to go full vertical the terminal velocities would be different.
Here is a paper that shows that while the life is zero on average it is not zero in all stages of flight.
It’s unlikely terminal velocity itself ever comes into play here, at least not in this way.
It would only come into play on the downward side of the trajectory. In that instance it would be best to have it travelling at the highest possible forward speed while having the slowest possible terminal velocity.
We are actually talking about the point of diminishing return here. If it happens to sort of coincide with terminal velocity here it is just that, coincidence.
The greatest factors will be how far it’s gone before reaching its peak and how high its peak is and how much forward speed is preserved on the downside. Speed, angle and momentum will determine all of that.
It is possible to reach a point of diminishing return though. Like with a lightly crumpled piece of paper, sometimes you can get it further with a gentle toss than with a hard pitch. The high initial speed creates high initial drag that bleeds of energy input sooner along the upward trajectory and then it freefalls
mostly straight down.
I think your friend is either using the term incorrectly or just using it to relate the concept more simply, or it’s coincidince since terminal velocity is a drag,mass,speed relationship with gravity as the energy input but this point of diminishing return has the bow as the source of energy input.
Dropping the arrow will never give you it’s terminal velocity in flight. Since it’s oriented forward with a downward slant as it falls in flight it’s terminal velocity would have to be determined in that position.
Of course as you stated , flex has a lot to do with it since it will burn energy in the form of an oscillating flight path.
Higher grain points could increase shaft flex.
…there’s a lot of factors here.
