How To Calculate Catapult Trajectory

I have an assignment where I have to build a catapult that is able to launch a bean bag any distance between 8-12 metres. My teacher will give me a random distance on the day it's due, and I have to be able to adjust the catapult in order for it to launch that distance. I'm having a bit of a problem with the calculations. How would I go about solving for the displacement of the spring needed to launch a certain distance?

Relevant Equations:

F=ma
W = Fcosθ(d)
Ek = 1/2mv^2
Eg = mgh
Fx = kx
Ee = 1/2kx^2
d = v1t + 1/2at^2

Attempting the Calculations:

mass of beanbag = 0.048kg
distance needed to travel = lets say 10m
Fx = 10N (using a spring force gauge)
x = 0.25m
launch angle = 50 degrees

Fx = kx
10 = k(0.25)
k = 40N/m

Ee = 1/2kx^2
= 1/2(40)(.25)^2
= 1.25

In the beginning it has elastic and gravitational potential energy so
Ee = Eg
1.25 = 0.048(9.8)h
h = 2.657m is this the max. height of the bean bag?

I'm thinking I need to solve for acceleration and velocity so that I can use
v2^2 = v1^2 = 2ad and that would be my distance, then I can work backwards using different distances and solve for Fx and x, but then I think I also need to use the launch angle in the calculations, and I don't know where to do that.
Thanks for looking at my post.

Just a tip, its probably going to be a lot easier to just experiment until you get it firing within that given range. Perhaps have a set spring/bungee and figure out how far you need to pull back the arm to have it hit different distances.

But if you were to actually calculate it you would need to know the spring stiffness 'k', then assuming its linear meaning it follows F=kx (which it may not) then the energy stored in the spring would be E=(1/2)kx^2. Next you would need to set up the equation for the angular kinetic energy of the throwing arm. In that equation the angular speed would be the unknown. Then set the stored spring energy equal to the kinetic energy of the arm at release to solve for the angular speed. From that point its a projectile motion problem.

Yeah I figured just experimenting would be the best route to take.

How would I know the value of k of the spring though? If I go to a store and buy a spring would it say what the stiffness is? Or could I just use the spring scale to measure the force of the spring, use the that to find the stiffness and then solve for energy stored?

Thanks again!

To find 'k' you would simply apply a known weight or force to it and measure how much it deforms. Do this a few different times with different weights and plot the results (force vs displacement) using something like excel. Then if its linear it will form a straight line (or close to it) with the slope equal to 'k'.

Also depending on what you use to build the catapult, you made need a fairly capable spring that may be harder to find. When I built one awhile back i made mine from wood, mainly 2x4's and just used a couple bungee cords until i got it firing a good distance.

You can do a nice experiment to figure out the constant. Get a set of known weights and hang the weights from the spring and measure the extension. Do this for a lot of different weights and graph the resulting data points on excel. Use a best fit curve and you can find a function that matches your spring. It should be approximately linear for relatively small extensions. If you set it up right, the slope of the line will be the spring constant. I did this experiment about a year ago. It is quite fun.

To find 'k' you would simply apply a known weight or force to it and measure how much it deforms. Do this a few different times with different weights and plot the results (force vs displacement) using something like excel. Then if its linear it will form a straight line (or close to it) with the slope equal to 'k'.

Also depending on what you use to build the catapult, you made need a fairly capable spring that may be harder to find. When I built one awhile back i made mine from wood, mainly 2x4's and just used a couple bungee cords until i got it firing a good distance.

Haha you must have posted this while I was typing my reply.

Haha you must have posted this while I was typing my reply.

Yeah i think so..i laughed when i saw your similar post almost instantly after mine.

Haha okay this makes much more sense now. It funny cause I actually did an investigation on Hooke's Law a couple weeks ago doing what both of you guys said to find the constant of a spring, but I totally forgot about it. Thank you guys so much!