# Thousand pound steel bow?

If a bow was made of steel and shaped like a Old Mongolian Bow with a draw weight of around 1000 lbs was made in proportion to a 9 foot tall humanoid (assume strength is enough to pull bow normally and the bow and arrow is roughly double regular size) what sort of force, speed, ect. would the arrow achieve? (Assume the arrow could survive these forces or adjust the arrow as needed)

• The range, speed, etc. depends on the mass of the arrow. For this question to be answerable, that information must be included. – Gryphon Jan 22 '18 at 19:02
• Ok i will edit to make this more clear – Christopher Void Jan 22 '18 at 19:04
• This sounds like a physics problem not a worldbuilding one. – sphennings Jan 22 '18 at 19:07
• I assumed it would be acceptable considering I need to know these things in order to realistically interact with it in my world – Christopher Void Jan 22 '18 at 19:11

# First of all, some conversions to make the physics easier:

500 lbs ~= 225 kg
43.2 in ~= 1.1 m


# Okay, now why these numbers?

Bows are basically springs, meaning that they follow Hooke's Law reasonably well. Hook's law states that $F=ks$ where $F$ is force, $k$ is a constant related to the spring and $s$ is how far you've stretched the string.

Now I'm going to make an assumption that's wrong, but will make all of this much easier. I'm going to assume that your bow begins from a state of rest, meaning that it starts at $s=0$. This is wrong, because the bowstring does impart some force onto the bow, creating some tension before you even begin to draw, but since this whole exercise is an estimate, I don't think this will affect our end results much.

In physics, work is defined as $W=F_{avg}s$. Here's where that assumption comes into the picture. Since the force exerted by the bow is linear, if it starts at a state of rest, then the average force is just half the maximum force. That means that the average force exerted while pulling back your hypothetical bow is 500 lbs, because that is half of 1000 lbs (the max force during draw).

Now we need $s$, or the draw length. This site tells us that the length from fingertip to finger tip divided by 2.5 should be the proper draw length. For our 9 foot tall humanoid, that's $(9*12)/2.5=43.2$.

Using the conversions above that's $F_{avg}=225$ and $s=1.1$.

# Now plug it in

Plugging our numbers into our formula above, we get $W=F_{avg}s=225*1.1=247.5\ Joules$. Not all of that is going to go into sending the arrow forward. A lot of it is wasted by making the arrow oscillate and some heating of the bow. According to this site, only 54.8% of the energy goes into the arrow. So, $E_{arrow}=W_{bow}*.548=247.5*.548=135.6\ Joules$.

# So how fast is it going finally?

Now kinetic energy is defined as $E_k=(1/2)mv^2$ so now all we need is the weight of the arrow. This guy wanted some help with the weight of his arrow, and his community was helpful enough to let him know it was ~375 grains... which is apparently a unit of weight? Anyway apparently that's about 24 grams. I'm going to go with a crude "let's double its size!" calculation. Doubling the size makes the volume increase by a factor of 8, and the weight by the same, so the weight of our extra large arrow is 194 grams. Note that double the size is 56 inches, which isn't too much longer than our draw length, sounds good to me!

Now throwing that in the equation, we get $135.6=(1/2)*.194*v^2$. Solving for $v$ gets us about 37 meters per second or about 80 mph!

• This shows the limits of mechanical energy storage. Even a very crude matchlock from the 1500's could provide about 1000J of energy to the projectile, an order of magnitude difference. And weapons like steel crossbows sized for normal humans could already deliver @ 200J of energy to the quarrel, so a longbow sized for a 9' tall human isn't even that impressive. Don't forget also that longbows need a lifetime of training and constant practice, while crossbows and firearms can be used effectively with minimal training. – Thucydides Jan 22 '18 at 23:08

So 1,000lb is grossly oversized.

Average draw weights for normal bows are in the 40-55lb range. You're talking about scaling up a human by 1.5x (6ft to 9ft), but increasing the arm strength by 20x. Even assuming a cube law this would indicate a more likely bow draw range of 3.5x or 150-200lb comparable to a crossbow. For reference the giant would need to dissipate this force when firing (equal and opposite), a thousand pounds force is one hell of a shove even for a giant bracing for the blow.

1000lb is not a bow it is a small ballista.

For reference these guys built a replica roman ballista of 700lb draw weight.

Ballista don't fire regular arrows, but more commonly stones, metal darts, or metal tipped spears. They also have a guide track for the projectile to follow, which your bow would lack making it very hard to aim or shoot.

Comparable draw weight ballistae can shoot several hundred meters when firing in a parabolic arc. They were known to pierce shields at range and were highly accurate and deadly.