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# How to calculate projectile distance of a trebuchet? Answered

Hi! After looking on the net for calculations to predict the distance of a projectile launched by a trebuchet, I've not found anything useful. There are many variables and each one can be changed to alter the projectile distance and velocity. I am looking for a simple way to calculate this without calculus.Thanks again.?

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I found some good web pages with highly detailed answers to predicting the range of a trebuchet.  A very simple model we have used in my Intro to Eng class just uses the mass of the projectile (m2), the mass of the counter weight (m1), and the height the counter weight falls (h):

Range (max) = 2 * (m1/m2) * h

Now the efficiency of the trebuchet will cause this model to be off by quite a bit.  But once you have a working trebuchet, we find this model works well when we vary m1, m2, or h.  We assume we have a take off angle of 45 degrees above the horizon.

This solution is based on the classic max range ballistics problem - 45 degree take off angle.  It also assumes converting all the potential energy of the counter weight to kinetic energy of the projectile.  That is why the efficiency issue comes up as a lot of energy is lost due to friction in the moving trebuchet.  If the projectile spins a lot then it will travel a shorter distance as the potential energy is split into kinetic and rotational energy.  Projectile shape and wind will also vary the results.

The students found this worked well enough for their lab work and it was lot of fun.  Good luck!

Intro to Eng with Physics?  How very interesting...
Thanks for the reply.  I just noticed it today.  The autonotify must not be working or getting caught in my spam folder.

Anyway, your solution appears VERY simple indeed!

I'll have to give a shot.  So far this is the best answer and I'll select it as such.

Check out http://www.algobeautytreb.com/ for a mac or windows app to do the predicting. Its a bit fiiddly - if you faff with one value then the others slip away from the optimal values. Otherwise works fine.

It should be possible to do some sort of maximisation calculations, if you knew all the formula.

Thanks for the reply.  The notification for new messages didn't notify me so I just found your message.

I'll check out your suggested website - sounds cool!

Alright... I don't want to sound mean or off-topic at all... but your username is bothering me. Since I'm assuming it's supposed to be pi to 5 significant figures, you have to take into consideration the 6th digit, which is a 9. Therefore, you are required to round up the 5 to a 6 making it 3.1416. Thank you.]

It is okay. :)
I did it on purpose to mess with the mathematically conscious. Of course, pi is an infinite, non-repeating decimal and all standard rules apply; I truncated instead of rounding.

Thank you for commenting Mr. Duck.

Work = force x distance

You (should) know the mass / weight of the counterbalance, and how far it travels during a shot. You should also know the weight / mass of the projectile, and how far it travels before release.

Assuming a spherical horse in a vacuum, they can state that work out = work in, and be able to calculate the KE of the projectile.

If they also know the launch angle of the projectile, that should convert quite easily into a predicted range.

When they fail to meet the predicted range, they can then work out why, which would be equally as educational as building the weapon in the first place.

Thanks Kiteman, I like and value simplicity! I think that is where I started and then began overthinking as well...I call it Analysis Paralysis (I read this somewhere once). Anyway thanks everyone in this forum for spending time considering and searching for an answer...

. You only need two factors - velocity and angle - to compute distance. You can ignore or fudge a lot of the variables Kiteman mentions, unless you need mm accuracy. The sling is likely to complicate things, but may boil down to a simple function. . Searching for "trebuchet +trajectory" turns up some interesting titles.

I thought Pi was after a way of predicting the range of an unbuilt trebuchet.

. Well, I was just trying to point him in the right direction and I didn't say it would be easy. :)
. Let's see if we can at least help him determine which variables are most important.
• mass of projectile. If it's reasonably dense and semi-aerodynamic (eg, rock, bowling ball), you can probably ignore air resistance. Should be able to find a reasonable approximation of effective area and drag online, if wanted.
• torque of machine. This should just be a function of counter-weight mass and lever lengths. Depending on construction methods, friction may or may not be very important. Can probably use a fudge factor of, say, 10-20% loss.
• lengths of rigid arm and sling. I can imagine that computing the launch velocity, with the sling attached, will be rather complicated. This is where a web search would come in handy. I don't think the rigidity of the arm(s) will have that much of an effect (unless they are very flexible). Along the same line, with a suitably strong sling, I don't think stretch will be that big of a deal.
. With that data, one should be able to compute launch velocity. I'm not the one, but someone should be able to figure it out.
. I have no idea how you would figure out the launch angle. Web search.

NachMa, Thanks for the excellent input...everyone in this forum is providing great advice!

Oh, I've been over-thinking this!

Work = force x distance

They (should) know the mass / weight of the counterbalance, and how far it travels during a shot. They should also know the weight / mss of the projectile, and how far it travels before release.

Assuming a spherical horse in a vacuum, they can state that work out = work in, and be able to calculate the KE of the projectile.

If they also know the launch angle of the projectile, that should convert quite easily into a predicted range.

When they fail to meet the predicted range, they can then work out why, which would be equally as educational as building the weapon in the first place.

(I'm going to re-post that as a proper answer - Pi is a physicist, he'll know the relevant equations).

True...kinematics equations. However, the initial velocity is unknown without a device for measuring or without calculating even some of the things that Kiteman mentions to arrive at a usable initial velocity. Thanks for the feedback...

Sorry, but I don't think such a tool exists. Off hand, you would need to factor in; Length of arm. Mass of arm Rigidity of arm Mass of counterweight Kind of attachement of counterweight (rigid, loose etc) Friction at pivot Angle arm turns during preparation Angle arm turns during launch Angle of hook holding ammunition Length of sling holding ammunition Angle of release hook for sling Elasticity of sling itself Mass of projectile Air resistance of projectile So you're best off just building it and seeing how far it goes, or comparing your design to the abilities of existing trebuchets of similar design.

All those variables definitely play integral parts in calculations. I am embarrassed to admit, I am a physics teacher and don't have a solution. My purpose for this is for my students to compete with their individual trebuchets and part of the contest would be to predict accurately their launch distance and accuracy in hitting a target. My only criteria for building the trebs was that they must be able to fit through the door of the classroom... I've been working on calculations on my spare time...which isn't much of the time so I was hoping to find someone would had already done it. BTW, my students loved the straw rocket design that I lifted from an instructable! There are a few stuck on high ceiling light fixtures that will be there forever...or until the lights come down.

http://www.algobeautytreb.com/trebmath35.pdf