Projectile Motion: Designing a Launching Device for Basketball Hoop

[Physics_wiz]

I have to make this launching device that shoots a projectile (anything I want) in a basketball hoop. I got everything figured out but just need someone to look over my work.

The device will consist of a spring inside a pipe. At the top of the spring (where the projectile will be) there will be a small flat piece of wood with a rope going through it and through the spring to the other end. The rope will come out of the other end and go under a pully supported on both sides and I will pull the rope to contract the spring and then let it go.

Here are my calculations:

the basketball hoop is about 10 ft high so that's about 3 m. I did
Vf^2-Vi^2=2ad where Vf = 0, Vi = initial vertical speed, a = 9.8 m/s^2, and d = 3m

I came up with Vi (initial vertical speed) = 7.668 m/s

I then used Vf-Vi = at to find the time in flight where Vf = 0, Vi = initial vertical speed, a = 9.8 m/s, and t = half of the flight time.

Total time ended up being about 1.56s

The device will be placed behind the foul line which is about 15 ft away from the basket or about 4.57m which means that the range of the device should be 2 * 4.57m = 9.14 m.

I then used V = d/t to find the horizontal initial velocity and came up with:
V = 9.14m/1.56s = 5.859 m/s

From the initial horizontal velocity and initial vertical velocity, I used vectors to find the initial velocity and the angle and came up with:

Vi = sqrt(Vih^2+Viv^2) = 9.65 m/s
angle = tan^(-1)(7.668/5.859) = 52.6 degrees.

So far, I found the angle at which the pipe should be placed and the initial velocity of the projectile, but to buy the string I have to find the string constant and here's what I did to find it:
F = kx.
KE (kinetic energy) = int(kx dx) = (kx^2)/2
KE = (1/2)(mv^2)

(kx^2)/2 = (mv^2)/2

k = (m*v^2)/(x^2) where m is the mass of the projectile, v is the initial velocity I came up with (9.65 m/s), and x is the displacement of the spring. I can control the displacement but it will probably be about .5 m so that gives me a spring constant of about 14.8996 Kg/s^2 (according to my last equation). The units are wrong so I probably did something wrong, anyone know where I went wrong?

 

[Physics_wiz]

Hmm, I played more with the units and figured out that Kg/s^2 is equal to N/m so I guess my units are not wrong. Are my numbers right though? Any tips or hints as to what might go wrong or numbers that are not realistic? Do you think the device will work?

 

[Kulaman]

well you have to realize that there are a large number of possible trajectories for your projectile, you also need to make sure that your projectile isn't going to reach 0 m/s vertical velocity when it reaches the 3 m height of the hoop. also, even once you get the calculations right, you have to remember friction inside the barrel, which will drop your initial velocity more than enough to make the device a failure. you've chosen a good way to approach the problem though, a cannon is more consistant than most, if not all, other possible launchers in this case

 

[Nenad]

Friction, work done by the drag force on the projectile. Make sure you have an extremely small projectile that is round to minimize air resistance. Remember, we are not in a vacum.

Regards,

Nenad