Catapults vs Guns

We were talking about guns, the velocities of their bullets and their ranges yesterday here:

On Guns, Missiles and Weapons of War vs A Peaceful Utopian World Outside Ours

https://scientificlogic.blogspot.com/2024/

 I mentioned that when I was a boy in school the only thing that looked like a gun for me to shoot stones at tins, objects and birds was a catapult.

 I used to make a catapult from a Y or U-shaped branch from a rubber tree, and as a boy I always looked for a V-shaped rubber tree branch in my hometown to make one.  I then cut-out old bicycle rubber tyres from a bicycle shop to power the stones like bullets.

Of course, these days we can easily buy a commercially made catapult from the market cheaply.

But I have always wondered about the speed of these stones, ball -bearings ejected from a catapult as they can be quite powerful to cause a small dent on a milk can.

 Of course, this sounds silly since catapults are not like guns that has a muzzle velocity and the range the stone ejected by a catapult will depend on the rubber used like a sling, its thickness, how long it is, and how far it is being stretched besides the weight or size of stones or pellet used.

 But let us assume on an average, how powerful it is, how high the stone can fly vertically upwards and the range horizontally.  

Catapults, while not as precise as modern firearms, can still demonstrate impressive ranges and projectile heights. The performance of a catapult depends on several factors, including the design (e.g., torsion, traction, or counterweight), the tension or counterweight used, and the weight of the projectile.

Factors influencing catapult performance depends on the design type such as torsion catapult that uses twisted ropes or sinew to store energy.

Then there is the traction catapult that I used to make that relies on hand power. There is also the counterweight catapult (Trebuchet) that uses a heavy counterweight to fling the projectile.

Tension or counterweight ones store the amount of energy in the tensioned element, or the weight of the counterweight significantly affects the power and range.

Projectile weight and lighter projectiles can be flung farther, while heavier projectiles might not travel as far but can cause more damage.

How far the projectile can travel depends on arm length. The longer arms can increase the range by providing greater leverage.

The average performance estimates of ranges of catapults depends on their type and design.  Counterweight catapults (trebuchet) in a well-constructed one can launch a projectile up to 300 meters (approximately 980 feet). Historically, larger trebuchets could achieve ranges of up to 500 meters (1,640 feet) or more.

Whereas torsion catapults can achieve ranges of about 100-200 meters (328-656 feet), depending on the tension and projectile weight.

The maximum height that a projectile from a catapult can reach is influenced by the launch angle, typically 45 degrees for maximum range. For a catapult with a range of 300 meters, the projectile could reach a maximum height of approximately 75 meters (246 feet).

Assuming optimal conditions, a well-constructed trebuchet can reach to a range of 300 meters

An example of calculation is:

For a launch angle of 45 degrees, the range (R) can be estimated using the equation for projectile motion:

R=   v₀²sin (2θ) / g

where v₀ is the initial velocity, θ is the launch angle, and g is the acceleration due to gravity (9.81 m/s²).

Rearranging for v ₀

v₀= √ (Rg / sin 2θ)

Plugging in the value, we have:

V₀ = √ (300 x 9.81 / sin 90 ⁰ )

Approx.  √ 2943

= ≈ 54.24 m/s 

Height Calculation:

The maximum height (H) is given by:

H = v²_u sin ² (θ) / 2 g

Using θ = 45 ⁰ and v _u = 54.2 m /s

H = (54.24)² sin ² (45 ⁰) / 2 X 9.81

= approx. 2943 x 0.5 / 19.62

= approx. 75 meters

Let us now compare a modern gun and a catapult.

Consider a scenario where we want to compare the efficiency of ancient catapults and modern firearms in terms of energy transfer to the projectile. How would we approach calculating the efficiency of both systems, and what factors would we consider in our analysis?

Let me provide some additional insights and clarifications

For a catapult, the energy source is typically the potential energy stored in the tension or torsion system (e.g., twisted ropes, counterweights). The efficiency of energy storage in the catapult mechanism lies in its energy transfer. The efficiency of transferring stored energy to the projectile depends on air resistance and friction, namely, losses due to air resistance and friction during launch.

In modern firearms the energy source is the chemical energy stored in gunpowder or other propellants. The energy release depends on the efficiency of converting chemical energy to kinetic energy of the bullet. In barrel design the barrel affects the efficiency of energy transfer (e.g., minimizing energy losses due to friction and heat).

We also need to consider air resistance and friction.  Loss of energy is due to air resistance and friction as the bullet travels through the barrel and air.

Let’s clarify the energy calculation for a firearm:

The gunpowder energy is 3 MJ / kg = 3000 J/g. The bullet mass (example) is 10 g = 0.01 kg. The gunpowder mass (example) is 30 g = 0.03 kg

Total energy from gunpowder:

Energy =3000 J / g ×30 g =90000 J

Assuming 100% energy transfer to the bullet (which is idealized):

KE = ½ mv²

90000 J = ½ x 0.01 kg x v ²

V² = 90000 x 2 / 0.01

V ² = 18000000

V = √ 18000000

Approx. 4242 m /s

This velocity is extremely high and not realistic due to inefficiencies. A more realistic velocity for bullets is in the range of 300 – 900 m /s

Air resistance, energy loss on conversion, barrel size or design were not considered. We assume 100 % efficiency energy conversion, and the projectile was in a vacuum. 

Readers may be wondering why I go into these physical sciences about catapults, guns and missiles?  I have also been interested in the speeds and power of projectiles. When I was a boy, calculation on speed and range of projectiles like bullets, missiles, slingshots and catapults was beyond me. Today having studied physics and mathematics at university degree level, this is not a problem. I am more interested in astronomy, physics and mathematics than in biological subjects like anatomy, physiology, biochemistry, medicine, nutrition, microbiology, molecular biology, genetics, immunology or in analytical chemistry and evolutionary biology. But I can tolerate forensic science and toxicology. 

 

ju-boo lim