# Trebuchet Physics: An optimization problem

1. Aug 6, 2004

### Nicholas

Let me present to you a problem of great diffculty.

For those who don't know what a trebuchet is, it is a medieval catapult which uses a counterweight (CW) to hurl heavy projectiles (P).

Although the art of building these siege weapons dates back to before the inception of calculus, optimizing the machine can prove very difficult even with modern tools.

The trebuchet uses moment arms to increase the acceleration caused by the falling CW. The energy is transfered to the projectile (which is much lighter) by use of the main arm and a sling. While the main arm (a) is longer than the counterweight arm (b) to increase the rotational velocity. In addition, the sling provides another radius (c) which further boosts the velocity of the projectile.

Example:
http://www.geocities.com/boeclan/trebuchet.bmp
*note if link does not work, copy and paste it into address bar.

The mechanics behind it are simple but the math is very complicated.
I challenge you to optimize the velocity of the projectile upon release by the trebuchet.

Once the arm (total arm length = a+b) has extended beyond its release point, the slip ring attached to the sling can no longer hold on to the release pin. This causes the sling to open and extended, hurling the ammunition.

To optimize its release velocity one must optimize the ratio between a, b, and c. The ratio between a and b determine how much faster the tip of the main arm rotates in relation to the counterweight arm (b). C determines how long the CW can free fall before having to accelerate the projectile. It is also another radius which changes the velocity.

Furthermore, the total acceleration time is determined by the distance CW can fall before it reaches its apex. Since the CW is free to rotate, assume it follows a completely straight path untill release.
Keep in mind that this device is gravity powered. If the ratio between a is not long enough compared to b, it won't achieve peak velocity, whereas if it is too long, it will apply more torque than the CW, causing a misfire.

Write the equation descibing projectile velocity in terms of a, b, c and two arbitrary masses (CW and P for projectile). Optimize solely on the ratios between a, b, and c.