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I have been looking at trebuchet designs lately, and I have noticed that most, if not all, have a sling attached to them. Without such a sling, the machine would be a catapult. In terms of the speed and energy of a launched projectile, what is the general difference between a catapult and trebuchet? Would trebuchet projectiles have extra centripetal acceleration due to the sling?

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The sling is not what makes it a trebuchet instead of a catapult. Catapults get energy from tension (usually torsion) and the arm hits a stop near the top of its arc to release the projectile. Trebuchets get their energy from gravity. The arm is off-center on an axle with the longer "launch" end (with or without a sling) holding the projectile and the shorter "load" end (locked in the "up" position before launch) loaded with something much heavier. When the load is dropped, the launch arm comes around at a much faster velocity due to its increased length. Think of it sort of like the tip of a bull whip can reach 900 mph, but your hand can not. It's just taking advantage of fulcrum points and counter weights to do so. The sling adds to that by extending the launch arm without increasing its weight by much while also improving accuracy. Throw a baseball with your hand cupped like a spoon and then throw it with your fingers spread to get max grip and it's easy to see which is more accurate.

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Let's say you have a catapult that can be turned into a trebuchet by attaching a sling to the end of the projecting rod. And let's say that it always launches the object with its projecting rod moving at some particular angular velocity instantaneously before it is stopped.

Adding the sling has some benefits in terms of launching something at an enemy. The sling tends to launch the object at a more horizontal angle which means it will plow a line through the target rather than dropping on it from above, this increases the damage output of the object, but also makes it much easier to aim. Additionally, the sling doesn't lose a lot of energy to friction, so this is an efficient way to redirect your projectile's momentum towards what you're trying to hit.

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Both operate on the same principle: the velocity of a point on a rigid body undergoing rotation is proportional to its distance from the pivot.

The sling is simply an ingenious way to extend the distance of the the projectile from the pivot without extending the rigid arm. As the trebuchet arm moves in an arc, the sling exerts a centripetal force on the projectile and is therefore pulled taut. The radius of the projectile's trajectory is now the length of the rigid arm plus a component of the length of the sling. For a given angular velocity, this increases the speed the projectile is travelling at.

Note: From further reading, it turns out this is a gross oversimplification of the kinematics of trebuchet launches. Here is a much more in depth analysis: http://www.real-world-physics-problems.com/trebuchet-physics.html

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Well ultimately the range depends on launch velocity. This is a combination of acceleration (applied force) and time or distance (during acceleration.)

The trebuchet seems to give a much longer "casting stroke" , so reaches a greater launch velocity. Very clever gadgets.

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