Optimizing Your Catapult: How to Calibrate for Maximum Distance and Accuracy?

[muddy_waters]

My current physics project is to design a catapult that will launch a marble to a given distance (between 2-3m). I have successfully made one out of wood, using a bungee cord to snap the arm back after being pulled on.

My question is how do I calibrate this properly? Our teacher said we need to use projectile motion equations and energy equations. I am not entirely sure how to do this so can someone help me please?

 

[LawrenceC]

You use the energy equations to determine the velocity of the projectile as it leaves the sling. Once it leaves you use the projectile motion equations to determine how far it will travel.

 

[muddy_waters]

I thought of that, but the question arises as to how to fire the marble at that specific velocity?

 

[dikmikkel]

How about measuring your spring constant(by a Newtonmeter) and you can calc. the spring energy: 1/2kx^2
Then you can calulate energies set the zero-point for potential energy when the catapult is loaded and then use energy conservation:
$E_{initial} = 1/2kx^2\\ E_{final} = mgy + 1/2mv^2 = 1/2kx^2\\ 1/2mv^2 = 1/2kx^2 -mgy \\ v = \sqrt{\dfrac{k}{m}x^2-2gy}$

 

[muddy_waters]

Actually nevermind my previous post. Would I use the following equation:
Elastic energy = gravitational potential + kinetic energy
Ee = Eg + Ek

If so how would I calculate the elastic constant for my specific bungee cord?

 

[muddy_waters]

dikmikkel, I do not have a Newtonmeter. Are there any alternative methods to calculating the coefficient ?

 

[dikmikkel]

You take an object which you know the weight, let it hang free and see how long it stretches the bungee and the gravity is the only force:
$mg - k\Delta x = 0 \\ k = mg/\Delta x$
So you would need to measure the initial length of the bungee.

 

[muddy_waters]

Ok so once I have the coefficient, I will use your derived equation to solve for velocity. Then how do I find the specific range? would I use this equation:

ΔDx = (V1sin2θ)/g

 

[dikmikkel]

You would use your eq.s for projectile motion:
$x = v_0 \cos(\theta)t \\ y = y_0 + \sin(\theta) t -1/2gt^2$
Where y0 is the height of the catapult just before shooting. Now the task is to eliminate the time from the above i.e. isolate the time in x and insert into y. then you would solve your quadratic equation for x with y = 0. Can you see why?

 

[muddy_waters]

Also in your derived equation, the velocity you solved for is initial right? Just making sure

 

[muddy_waters]

Sorry I don't follow, could you explain why I need the y as well as the x?

 

[dikmikkel]

oops i made a booboo:
$y = y_0 + \sin(\theta) v_0 t -1/2gt^2$
But yeah that is right the v0 is the velocity from #4.
Edit: You need the time to be eliminated, because you don't know how many seconds your marble will fly.
So the only way is to define the ground as y=0 and therefore the y is needed as well as the x.

 

[dikmikkel]

You don't need the y it is only interesting when set to zero.

 

[muddy_waters]

I understand what you're saying but shouldn't the ground be -0.48m? The distance from the ground to where the marble is launched on my catapult is 48cm above ground level. Or is this distance insignificant enough to rule out?

 

[dikmikkel]

No it is not. But i choose ground to be y=0 hence the y0 term in the expression for y.
Edit: With your suggestion the y_0 should be zero, and we would get the same result.

 

[muddy_waters]

So setting y0 = -0.48 I derived the following equation:
y = -0.48 + xsinθ/vcosθ - 1/2g(x/vcosθ)^2
Is this correct? Now all I have to do is set the equation to zero, and plug in the x which is the given distance, and solve for v. Then from there I work backwards and use the energy eq'ns to solve for how much to pull it back?

Also is θ the angle at which the arm is to the ground?

 

[dikmikkel]

You probably want the y0 to be positive 0.48 else you have to make some odd changes.
Make a sketch with your definition of the coordinate system and see.
the theta you got right.
Hmm you could do that, or you could derive a equation which only need to take the desired firing range x? It could be done from our discussion and eq.s.
But i get the same eq. except from the sign on y0

 

[dikmikkel]

When doing that you would insert the v from #4 as v0 and arrive at the following result:
$0 = y_0 + \dfrac{\sin(\theta)}{\cos(\theta)} x -1/2 g x^2\dfrac{1}{\cos(\theta)^2 (\dfrac{k}{m}\Delta x^2-2gy)}$
Which you would need to solve for delta x :)
Edit: That is good physics: Obtain a finished equation and plug in your numbers at last then you could also optimize it by varying the catapult height and so on.
EditEdit: The y is now the catapult height. sorry.

 

[dikmikkel]

So as a final result i get (by sin/cos=tan):
$\Delta x = \sqrt{\dfrac{gx^2m}{2k(y_0+\tan(\theta ) x)\cos(\theta )^2}+\dfrac{2gym}{k}}$

 

[muddy_waters]

Going back to the energy equations, I think you missed an 'm'. It's supposed to say -2mgy right? And for the y, would that be the distance above ground at the end? So wouldn't that be 0, making Eg negligible?

 

[dikmikkel]

#20 no you divide all the terms by m. And the y is the height of the stretched catapult e.g. when the marble fires

 

[muddy_waters]

Ok so this delta x you solved for is it the total distance fired or the total distance the bungee needs to be stretched?

 

[dikmikkel]

Total need to be stretched and the x is the total fired dist.

 

[muddy_waters]

Oh my God thank you so much. I really appreciate your help! :)

 

[dikmikkel]

Np. but the y and y0 are the same i just realized, and i like the problem

 

[muddy_waters]

Actually on #18 did you forget to put the velocity in the first part? 0 = xtanθ/v -1/2g...

 

[dikmikkel]

No originally it looked like $v_0\tan(\theta) x / v_0$ so it is not forgotten.

 

[muddy_waters]

Oh right, I forgot to cancel the v0 when using the projectile equations. Thanks again!

 

[dikmikkel]

Np. Can you see the advantage of doing your formulas by youself? Now you could design the optimal catapult for your needs and constraints(is that the proper english word?)