How Much Counter Weight is Needed for a Trebuchet to Hit a Specific Distance?

[Water Man]

Homework Statement

We need to find the mass of the counter weight needed to make our projectile go a certain distance.

Height = 22.5cm (The height of arm off the ground)
Distance needed to travel = 1m/100cm and and 50cm
Angle above horizontal = 32 degrees
Projectile mass (small ball) = 125 grams
Counter Weight mass = WHAT WE ARE CHANGING


We don't know velocity or anything else like that. How can we find out how much mass to put on the counter weight to make it travel a certain distance? Also please give equations/units. This is really bugging me. I tried many things and formulas but nothing seems to be working for me.


What I've tried

I tried using a lot of formulas but so far all my stuff is theoretical. I know I can find velocity using distance/time but I can't do it like that because I still haven't gotten it to launch that far. I have stumbled upon lots of formulas and tried some from this PDF:

[PLAIN]http://www.algobeautytreb.com/trebmath35.pdf

[/PLAIN]

But a lot of the stuff is too advanced and isn't specifying units. The one formula I did try was this: Range = 2 * (Counter weight mass/projectile mass) * h.

Height = the height the counter weight falls to.


However, the result's I'm getting seem very off.

 

[RTW69]

The article you cite is a useful resource.

I assume the 32 degree angle is the angle the projectile leaves the Trebuchet, call it "A". For projectile motion the Range,R = (2*Vo^2*sin(A)*Cos(A))/g

Solve for Vo

The KE of the projectile= M2*Vo^2/2 where M2 is the projectile mass

The PE of the counter weight is M1*g*h where M1 is the counter weight mass

Theoretically all the PE of the counter weight goes to the KE of the projectile. Setting them equal gives.

M1=(M2*Vo^2)/(2*g*h)

The article also describes a method of calabrating the Trebuchet by determining a range efficiency, eff. You need to fire the Trebuchet and measure the range for a given M1.

A new M1=(M2*R)/(2*h*eff)

 

[kerimek]

I'm not sure if I'm using the wrong units or if I'm missing something. I also tried using the formula for potential energy (PE = mgh) and equating it to the kinetic energy (KE = 1/2mv^2) at the point of release, but that also didn't seem to give accurate results.

I would suggest approaching this problem by using the principles of projectile motion and energy conservation. Firstly, we can use the equation for range of a projectile, which is R = (v^2 * sin2θ)/g, where R is the range, v is the initial velocity, θ is the angle above horizontal, and g is the acceleration due to gravity.

Since we are given the distance needed to travel (1m/100cm and 50cm), we can plug that into the equation and solve for v. This will give us the initial velocity needed for the projectile to travel the desired distance.

Next, we can use the equation for kinetic energy, KE = 1/2mv^2, to find the initial kinetic energy of the projectile. We know the mass of the projectile (125 grams) and we just solved for v, so we can calculate this value.

Now, we can use the principle of energy conservation to determine the mass of the counter weight needed. The total energy at the point of release will be equal to the sum of potential energy and kinetic energy. So, we can set up the equation PE + KE = mgh + 1/2mv^2, where m is the mass of the counter weight, g is the acceleration due to gravity, and h is the height of the arm off the ground (22.5cm).

Solving for m, we will get the mass of the counter weight needed to achieve the desired range. It is important to make sure all units are consistent throughout the calculations.

I understand that this problem may be challenging, but it is important to keep in mind that theoretical calculations may not always match with real-world results due to factors such as air resistance and friction. It would be beneficial to also conduct some experiments and make adjustments as needed to fine-tune the calculations.

In conclusion, by using the principles of projectile motion and energy conservation, we can determine the mass of the counter weight needed to make the projectile travel a certain distance. It is important to carefully consider all factors and ensure that units are consistent throughout the calculations. I hope this