Physics of Trebuchet: Argument with a Friend

[Overlord12345]

I'm in an argument with a friend of mine. He thinks that a trebuchet's maximum range is directly related to the mass of it's counter weight claiming that all the energy from the counterwiehgt would be transferred to the projectile. I think that it is logarithmicly related because the counterweight cannot accelerate past 9.8m/s/s; therefore, the projectile can only be launched at a maximum speed determined by the Height that the counterweight is dropped. We are argueing theoretically so we are removing friction and flex/damage to the trebuchet in the situation. I would appreciate input inorder to put the argument to rest.

 

[Hurkyl]

All that matters here is the kinetic energy of the projectile, correct?

So, if you had a way to compute the amount of energy the Trebuchet can transfer into your projectile...

 

[Overlord12345]

But since the counterweight has a maximum velocity which it could be moveing at the point of launch due to the fact that it's acceleration CANNOT surpass 9.8 m/s/s, the projectile has a maximum launch velocity. Unless i am mistaken this would mean that the projectile has a limit to the distance that the projectile could fly.

 

[Hurkyl]

Hrm -- are you guys considering the limits of a specific trebuchet, or are you considering what it is theoreitcally possible for a trebuchet to achieve with a given counterweight at a given height?

 

[cesiumfrog]

You know exactly how much gravitational potential energy the counterweight will loose, it doesn't really matter how quickly it does so (lever.. the two ends don't accelerate at the same rate).

 

[Quaoar]

m_counterweight * g * h = 1/2 * m_object * v^2. To double the velocity of the projectile, you have to increase the mass of the counterweight by a factor of 4. Distance traveled will be directly proportional to the launch velocity of the object. So, all other things held constant, a trebuchet's maximum range doubles for every quadrupling of the counterweight.

 

[DLinkage]

Well, the counter weight is related, but so is the moment arm of the counter weight, and the sling design. In order to maximize the range you really need to maximize the velocity, and optimize the angle when the projectile is launched. The hinge where the sling attaches has velocity proportional to the net moment produced by the counter weight. However a counter weight too far out also increases the rotational inertia of the system. To get a quick angular acceleration it is desirable to move the counter weight outwards as the sling increases in rotational velocity. The idea here is that by accelerating the projectice faster from the start, we will get more of an effect through a smaller time frame and thus small angle of rotation. While the work done is proportional to the initial potential energy of the system. The way in which it is release depends highly onthe design. Essentially you want to get the cross product of all the angular velocities to line up in a single direction which maximizes the range. However, by design the optimal launch angle of a simple projectile (45 deg) may yield a low launch speed on a trebuchet. Poor design could yield maximum launch velocity at 90 degrees - which would be very bad if you think about it a couple seconds later... Thus some more optimal angle must exist between the two. This be range is in no way linearly proprtional to the moment, while terms in the energy expression may be, the additional intertial terms caused by motion will also create an effect which is entirely based upon the design

 

[Hurkyl]

You know exactly how much gravitational potential energy the counterweight will loose, it doesn't really matter how quickly it does so (lever.. the two ends don't accelerate at the same rate).

Don't forget that some of that GPE goes into the kinetic energy of the counterweight.

 

[cesiumfrog]

Don't forget that some of that GPE goes into the kinetic energy of the counterweight.

Not if it's designed well. Hence, you see "modern trebuches" with floating axles..

 

[Overlord12345]

You know exactly how much gravitational potential energy the counterweight will loose, it doesn't really matter how quickly it does so (lever.. the two ends don't accelerate at the same rate).

But the two ends accelerate at a ratio determined by the placement of the fulcrum and since that ration is constant for every fireing of a given trebuchet, wouldn't it have a maximum range determined by v2=(v1*v1 + 2(9.8)(h)). If v2 is the velocity of the counterweight at launch then the velocity of the projectile is v2 multiplied through the arm and the sling. This would mean that it couldn't fire past that distance. Right?

 

[cesiumfrog]

Hmm.. . the concept of a trebuchet is just a device to effectively convert one object's gravitational potential into another object's kinetic energy. The efficiency of a particular trebuchet depends on the design, projectile and counterweight.

I think OP is correct in that, for a given trebuche+projectile+counterweight, the range (a function of firing velocity) does not only depend on how far the counterweight can fall. Whereas the friend is correct that, given the most optimal choice of trebuchet+projectile, the height a particular counterweight can fall does determine the maximum range.