If I took two very different size steel balls, say 1/2" and a 2" ball. I shot them both at say 500 fps at a 45 degree angle. Would they travel approx the same distance?
In air, no. In a vacuum, yes.
Got it in one. You have to account for friction caused by air. These two balls have vastly different surface areas.
I don’t think it would be equal in a vacuum either. The two balls are acted upon by the same force, but:
F=M*A
therefore,
A=F/M
Since the two balls have different masses it is apparent that their acceleration will be different.
Am I missing something?
They’re acted on by different forces, though: they’re both moving at the same velocity (500 feet per second) even with the different masses, so the accelerations to get them there were different. Once they’re at that velocity and travelling, the only acceleration is gravity (in the vacuum). Note that they’ll hit with different forces (because of the M in your equations), but we’d expect that.
You’re right. I was thinking that they were fired with the same force.
ETA: Thanks.
I am interested in the balls flying through air not a vacuum. I am trying to come up with a strategy for building flight arrows for this years competition. I am limited to arrow length, and size of my fletchings. I am not limited to the weight of the arrow or the thickness of arrow or fletching beyond that they have to be made from natural materials. I have physical limitations in that they have to be stiff enough to withstand the acceleration forces of the bow.
Last year I succeeced in getting my arrow weight down about 75 grains in weight and still maintain good flight, this gave me about 30 fps gain in speed comming out of the bow. My distances fell flat even though I had greater speed and good flight.
My thoughts now are that I need to figure out the optimum shape for the arrrow and try to find a thinner fletching material. Would a torpedo shape possibly have an advantage over a long needle tip?A heavier arrow would lose less speed to drag, but a lighter arrow would have a greater initial velocity if shot from the same bow, so it’s not simple.
Thats exactly my delemna, my cut off point based on the present shape of my arrows seems to be about 250 grains. I need to somehow lower that by about 100 grains.
One of my close friends and competitors has been producing arrows that have been responsible for nearly all the records broken in the last few years. He is an engineer and builds his arrows like a split cane flyrod from bamboo. Very difficult process that very few of us have been able to master. He seems to feel the superior stiffness of his arrows is the key as well as a smaller diameter. Judging by his successs I hesitate to say he is wrong but I am more inclined to think it may be the shape of his arrow in addition to his superior stiffness. I am able to equal his clean flight comming out of the bow, I can equal his arrow weight and diameter yet his arrows will travel up to about 40% further than mine even when I am shooting his arrows side by side with my arrows. They are simply traveling better. His fletches were lighter and thinner which I know helped a lot and I can make that change this year but I still think his arrow shape was the biggest factor.
Historicaly we have made our arrows in a barrel shape while still trying to maintain a slight forward weight ballance. I am thinking now a straight taper from tip to tail with a rounded blunt nose like a bullet might work best? Any thoughts on this?
A teardrop shape if that’s what you mean. It should result in much less drag. The shape where it attaches to the shaft can be critical, possibly tapering far back on the shaft. Also, shifting the CG forward should allow you to use much smaller fletching.
Your OP question asked about two projectiles with different mass but same initial speed. That’s a bit different from your arrow situation, in which arrows of different mass will be launched with the same initial kinetic energy, as determined by how far back you pull the bow. For a given amount of kinetic energy, the heavier arrow will have less initial velocity. That means less aerodynamic drag right off of the bow, but that doesn’t necessarily mean it will go farther: If you can launch a 30-gram arrow 100 meters downrange, it’s unlikely a 30-kilogram arrow will go very far at all.
I think this will be a situation where you need to do some testing and careful record-keeping to arrive at an optimal solution.
Maintaining Forwaard center of gravity is one the arrow makers biggest challenges. The further forward the weight the better flight we get, but it also requires more stiffness to resist the acceleration forces so the weight starts going up. Most of us have settled for about 5% FOC to maintain lower arrow weights where from a standpoint of flight you cannot have too much FOC.
Seems difficult to predict the performance. I think your friend is on to something with the stiffness though, I’ve seen the slo-mo videos of arrows bending and wobbling when released and in the air. The loss there is going to be a huge factor in total distance. Do you know the velocity of the arrow when released? From that you might be able to find out the ideal shape of the head.
Just curious about this, do you have a way to make a hollow shaft? A hollow shaft with a greater diameter of the same weight might give you something stiffer (not to mention the compressed air in the shaft prank we discussed before).
Thse are primitive all wood bows as the velocity is at best about 260 fps. The diameter of the arrows at the center is right around 5/16’ never below 1/4". I am at the point now where I am thinking I may have to give up a little to him in arrow weight if I can make it up somehow with a better shape. I just can’t seem to find the tipping point weight I need to be concentrating at.
Ok, that’s about 177mph if my maths are correct. Pretty good speed, not what you normally need a large bulbous nose on an airfoil for. You’re probably looking for something more streamlined, but still with a rounded nose, not a point. If you can make the head hollow to control the weight the head could taper far back on the shaft increasing the stiffness in the forward portion at least. Also some tiny protruberances a little forward of the fletches might act as turbulence generators making tiny fletches more effective. This is all getting kind of out there for designing an arrow though, but the ideal shape is something that tapers from the widest part of the head all the way back to the end of the arrow, with the widest portion as much as 40% back from the tip of the arrow, and the width there as low as 12% of the total arrow length. That’s just to reduce drag, but a hollow shaft like that might end up pretty stiff.
For a given shape and dimensions but different mass, given two perfectly rigid arrows both fired from the same bow, the lighter one will always travel further. If this is not your empirical result, then one of the initial assumptions must be incorrect. My gut inclination would be that your friend is right, and that the lack of rigidity is the most significant factor.
That seems unlikely to be always true. If the two projectiles start with the same speed, the heavier one will go farther. Of course, the lighter one will have a higher initial speed. But clearly there is some upper limit to how fast a bowstring can move, and so when you get light enough, you must hit a point of diminishing returns. Surely there is some cutoff where the speed advantage of a light projectile does not exceed the momentum advantage of a heavier projectile.
I find a small rocket engine helps significantly.
Hm, that’d be true. The point of diminishing returns would probably be at about the point where the mass of the arrow were comparable to the mass of the bowstring, though, which I imagine we’re still far above, even with the lightweight arrows.
And the same energy will mean that the lighter arrow will have a higher initial speed, and even though it’ll feel greater drag as a result, it’ll still at any given moment of time always have a higher speed than the heavy one.
The string of the bow plays a major part in the speed and efficiency. But their is no point at which a lighter arrow will not come out faster. Modern class flight bows are shooting arrows in the 50 grain range. These are the ones going out a mile or so. They are made from solid carbon rods.
Every bow has a unique virtual mass to it. The virtual mass is the difference between 100% efficiency and it's actual efficiency which might be around 80%.
Flight bows will favor designs that favor efficiency over stored energy as lighter arrows require a lower virtual mass bow to maintain some efficiency.