Elementary Speargun Physics

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What do we want from a gun?

We want a gun to pack a good punch accurately, at a good distance from the gun, while keeping various constraints in mind. Constraints include:

What do we mean by a good punch? We ideally want the spear to go right through a fish, so that the barb or slip-tip can open on the other side of the fish, while possibly passing through tough skin, scales, gill cover, or bone on its way. Let us assume that the spear tip has been designed to do this as well as possible (and is sharp), what else can we do to improve? Obviously the spear must go faster and/or be heavier. Let's assume that the speed at which the spear is travelling through the fish does not influence the force which the spear needs to pass through the fish. The work (or energy) that the spear needs to go right through is the force encountered by the spear integrated over the distance through the fish. (If the force were constant this would simply be the force times the thickness. If the force varies on its path through the fish, you draw a graph of this force through the fish and get the area under this graph). This means what we want from the spear when it reaches the fish is energy. The energy is measured in J (Joule) or Nm (Newton meters) and is calculated for a spear in motion as:

E = 1/2 m v2

where m is the spear mass and v the spear speed.


Where do we get this energy? From the rubbers of course. To get the energy that the rubbers produce, you integrate the force of the rubbers over the distance that they pull along the gun barrel. So if the rubbers are stiffer, or the barrel longer, the spear gets more energy.

Unfortunately the picture is not complete. The spear has to pass through water on its way to the fish. The spear loses a lot of energy even by passing through a few meters of water. In the water the resistance against the spear depends very much on its speed. The higher the speed, the higher the resistance. (Put your hand out of a car window to feel this.)

We can see that for a given barrel and set of rubbers, the muzzle energy of the gun is constant. What can we do to improve? We use a heavier spear. The muzzle energy mv2/2 stays constant, so with mass m greater, speed v gets less and the water resistance over the whole journey to the fish gets less, so that we end up with more energy to penetrate the fish. Obviously we cannot make m too much, because then the spear will travel so slowly that the fish may dodge and it will sink too far (if shooting horizontally) before reaching the fish. If you are happy with this explanation, don't read further. If you are wondering whether increasing the spear diameter when increasing spear mass won't cause more drag etc., read on.

A model of water resistance r, given spear speed v can be approximated by:

r = -k1v -k2v2

where k1 and k2 are constants that depend on the spear shape and dimensions. The negative signs are because the resistance force is in the opposite direction to the direction of the spear's travel. The k1v term which dominates when the speed is low, is due to the viscosity of the water. The k2v2 term which dominates when the speed is high, is due to the spear having to replace the water in its way against the water's inertia. We are mainly interested in the high velocity part of the spear trajectory, so we'll ignore the first term. We can solve the implied differential equation to get an expression for the speed of the spear as a function of the distance travelled. We then get the water resistance as a function of the distance travelled, using the same formula again. We now integrate the resistance over the distance travelled to get the energy loss as a function of distance. The friction constant k2 depends mostly on the wetted area of the spear. A heavier spear of the same material has to be thicker, which will cause a greater wetted area. Fortunately the thicker spear still turns out to do better. The results of the above (vaguely described) calculations is an expression for the spear energy at distance x from the muzzle:

E(x) = E0 e-kx/(rd)

where E0 is the muzzle energy given by the rubbers, k is a suitable constant, d is the spear diameter and r is the density of the spear material. For a given spear, the energy decreases as the distance increases, as we all know. But if we take a thicker spear, the decrease happens over a longer range. We see that if we want to increase our range x, without sacrificing spear energy, we must increase spear diameter by the same factor. If you want to shoot effectively at 10m instead of 5m, you have to use a spear that has double the diameter, if you still want to use the same old rubbers! We also see that the spear length is not present in this formula, but my experience shows that longer spears of the same diameter work better. This is due to the following factors:

  1. The muzzle eneregy E0, for a longer spear will be better. The spear is heavier, so the rubbers move more slowly, so the rubbers themselves expend less energy on drag and on wasted kinetic energy after the spear has disengaged from them.
  2. We did not consider drag on the ends of the spear, on the barb and on the line.
  3. The constant k in the equation above is actually slightly dependent on the length (via the logarithm of the Reynolds number). It gets smaller (less drag) for longer spears.
I have written a speargun simulator that calculates the spear trajectory based on these priciples. I am going to translate it to Java and make it available on this site soon.

To sum up what we have achieved with our assumptions and calculations:

  1. Energy is what we need to penetrate fish.
  2. Muzzle energy depends on barrel length and rubber tension.
  3. Energy is lost through the water, but heavier slower spears lose less energy.

Why platinum spears? Platinum is almost three times as dense as steel! Lets say we want to use platinum to shoot twice as far - it turns out we need a spear which has only about half the mass of the original steel spear, but it still shoots twice as far! The lighter spear will of cource travel faster, elimintaing the problem we get when thick spears travel too slowly. Unfortunately I don't know any spearos who could afford even one platinum spear!

I have considered filling a steel tube with lead. However, lead is not that much denser than steel and in the end you will have a spear that bends very easily!

Also have a look at our freediving club site.

Back to Roller Gun page.

Email me: niko@bonitofreedive.org