There has been some research on speargun dynamics, biggest paper on it was by Spiller (he posted about it here some years ago) I think. It showed that one of the most important factors isn't just force (m*a) but rather the work done throughout the barrel. This charges the shaft with KE which is what fights drag and allows the shaft to maintain speed (and thus momentum) better. It's why rollers/inverts shoot more powerfully with half the rubbers (total kg on the shaft/mech) and can use lighter shafts. I don't use pneumatics but my understanding is that they're like rollers.
Other tests from Rob Allen (very heavy shafts) and another guy from Greece have been made but that's mainly for the acceleration curve. Maybe that's what you mean by chronograph? See pics
Rollers/inverts accelerate more slowly and the shaft spends much more time and distance under peak or near-peak force which allows for incredibly more energy charge. But because there is a speed limit (rubbers themselves contract at about 50 m/s without load) you can only barely squeeze out some more velocity... but the energy can go up! This is the biggest takeaway and it's rather counter-intuitive. So you can get shafts with very similar maximum speeds but very large energy differences. Momentum also scales with KE
p=√(2m⋅KE )
This is just one of the various equations that are in play. Here are a few others
dv/dt = - (k / m) v²
(1/2) m v₀² = ∫₀ᴰ (k v(x)²) dx
(1/2) * ρ * C_d * A * v²
And so on, for calculating different things. I didn't expect it to be as complex when getting into it but it's actually easy to understand after you get the whole picture. There is no paper (or study) of these out there yet as far as I know. Spiller was the only one to get into KE up until the muzzle. And up until then -2021- nobody really had any idea how to calculate it, people just assumed velocity is the key (it's not).
Then there are other factors like less jerk which results in smaller energy loss during firing.
For shaft mass there is an approximate table. See pic
In the Med, 370gr are enough to hit a fast-moving 3-4kg fish that's 5m away from the tip. Heavier shafts fired with the same power will travel farther away but will reach the target less fast. So it really depends on what you're hunting and from what distance.
Other tests from Rob Allen (very heavy shafts) and another guy from Greece have been made but that's mainly for the acceleration curve. Maybe that's what you mean by chronograph? See pics
Rollers/inverts accelerate more slowly and the shaft spends much more time and distance under peak or near-peak force which allows for incredibly more energy charge. But because there is a speed limit (rubbers themselves contract at about 50 m/s without load) you can only barely squeeze out some more velocity... but the energy can go up! This is the biggest takeaway and it's rather counter-intuitive. So you can get shafts with very similar maximum speeds but very large energy differences. Momentum also scales with KE
p=√(2m⋅KE )
This is just one of the various equations that are in play. Here are a few others
dv/dt = - (k / m) v²
(1/2) m v₀² = ∫₀ᴰ (k v(x)²) dx
(1/2) * ρ * C_d * A * v²
And so on, for calculating different things. I didn't expect it to be as complex when getting into it but it's actually easy to understand after you get the whole picture. There is no paper (or study) of these out there yet as far as I know. Spiller was the only one to get into KE up until the muzzle. And up until then -2021- nobody really had any idea how to calculate it, people just assumed velocity is the key (it's not).
Then there are other factors like less jerk which results in smaller energy loss during firing.
For shaft mass there is an approximate table. See pic
In the Med, 370gr are enough to hit a fast-moving 3-4kg fish that's 5m away from the tip. Heavier shafts fired with the same power will travel farther away but will reach the target less fast. So it really depends on what you're hunting and from what distance.