Solving Projectile Motion Equation for Distance

[roldy]

Could someone help me come up with the distance equation for a projectile that is launched at an angle and initially at a height. I need figure out a relationship between distance, alpha, and beta. The final equation should contain only variables d, alpha, and beta.

The knowns:

1. The mass of the projectile is .021 kg
2. The distance from the pivot point of the pole to the ground is .2 meters

The unknowns:

1. The velocity of the projectile when the pole is at angle beta
2. The final distance
3. The angles beta and alpha

Here's what I've tried to do:

I started with the conservation of energy.
KE1=0
PE1=mgh₁
KE2=1/2mV₂²
PE2=mgh₂

I think I have solved this problem. I will scan in my work sometime and would like to see if I'm on the right track. If anyone could help with this problem it would be awesome. This problem is a theoretical problem for a design project I'm working on.

 

[meltingwax]

I think you need to be specific as to how the system actually works. Is the long pole rotating clockwise while applying a force on the projectile from angle alpha to beta? If this is the situation, then you can get an equation involving F, alpha, beta, and d. You would not be able to get an equation with just alpha, beta, and d (d is completely dependent on the force applied over the range theta=alpha to theta=beta).

If you had the force as a function of theta, then you could integrate over theta=alpha to theta=beta to get the work done, and then from there you can deduce the kinetic energy, velocity direction, and position upon release, and it becomes elementary.

 

[roldy]

Sorry about that, the long pole does rotate clockwise. The force that is applied to this system is done by a spring that will be attached from the pole to the front of the setup. I know that as the pole will be rotated back initially the spring will get stretched around the shaft a little but I'm neglecting that. In the equations I have derived, I will optimize the angles alpha, beta and the energy of the spring using the excel solver tool. Attached is what I think the equation looks like.

 

[roldy]

So I see everyone gave up.