Copy of Chris catapult

364 days ago by roosh

#NOTE: NEED TO CHANGE M (MASS OF BUNGEE TO HALF OF ITS VALUE) #m of everything (shuttle and ball) = .039, mball = .08, m = #wnet = kf - k0 #Wcord - Wgrav bungee = Kf #Wspring - M*g*h = 1/6*M*vf^2 + 1/2* m*vf^2 
       
#Work Wcord, Mcord,g,hrel, h0 ,mball,vf, m, dist, theta = var('Wcord, Mcord,g,hrel, h0,mball,vf, m, dist, theta') q = solve(Wcord - (mball)*g*(dist*sin(theta)) == 1/6*Mcord*vf^2 + 1/2* m*vf^2, vf) view(q) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\mbox{vf} = -\sqrt{-\frac{\mbox{dist} g \mbox{mball} \sin\left(\theta\right)}{\mbox{Mcord} + 3 \, m} + \frac{\mbox{Wcord}}{\mbox{Mcord} + 3 \, m}} \sqrt{6}, \mbox{vf} = \sqrt{-\frac{\mbox{dist} g \mbox{mball} \sin\left(\theta\right)}{\mbox{Mcord} + 3 \, m} + \frac{\mbox{Wcord}}{\mbox{Mcord} + 3 \, m}} \sqrt{6}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\mbox{vf} = -\sqrt{-\frac{\mbox{dist} g \mbox{mball} \sin\left(\theta\right)}{\mbox{Mcord} + 3 \, m} + \frac{\mbox{Wcord}}{\mbox{Mcord} + 3 \, m}} \sqrt{6}, \mbox{vf} = \sqrt{-\frac{\mbox{dist} g \mbox{mball} \sin\left(\theta\right)}{\mbox{Mcord} + 3 \, m} + \frac{\mbox{Wcord}}{\mbox{Mcord} + 3 \, m}} \sqrt{6}\right]
#Work in ugly form print(q) 
        
[
vf == -sqrt(-dist*g*mball*sin(theta)/(Mcord + 3*m) + Wcord/(Mcord +
3*m))*sqrt(6),
vf == sqrt(-dist*g*mball*sin(theta)/(Mcord + 3*m) + Wcord/(Mcord +
3*m))*sqrt(6)
]
[
vf == -sqrt(-dist*g*mball*sin(theta)/(Mcord + 3*m) + Wcord/(Mcord + 3*m))*sqrt(6),
vf == sqrt(-dist*g*mball*sin(theta)/(Mcord + 3*m) + Wcord/(Mcord + 3*m))*sqrt(6)
]
#kinematics #note that vf is subject to change based on previous equation vf = sqrt(-dist*g*mball*sin(theta)/(Mcord + 3*m) + Wcord/(Mcord + 3*m))*sqrt(6) xf, t = var('xf, t') eq1= xf == 0 + vf*cos(theta) *t eq2 = 0 == hrel + vf*sin(theta)*t-1/2*g*t^2 s = solve([eq1,eq2],xf,t) print(" ") print(" ") view(s) print(" ") print(" ") 
        
 
 
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[\mbox{xf} = -\frac{2 \, {\left(3 \, \mbox{dist} g \mbox{mball} \sin\left(\theta\right)^{2} \cos\left(\theta\right) - 3 \, \mbox{Wcord} \sin\left(\theta\right) \cos\left(\theta\right) + \sqrt{3 \, \mbox{dist}^{2} g^{2} \mbox{mball}^{2} \sin\left(\theta\right)^{4} + 3 \, \mbox{Wcord}^{2} \sin\left(\theta\right)^{2} - {\left(\mbox{Mcord} \mbox{dist} g^{2} \mbox{hrel} \mbox{mball} + 3 \, \mbox{dist} g^{2} \mbox{hrel} m \mbox{mball}\right)} \sin\left(\theta\right) - {\left(6 \, \mbox{dist} g \mbox{mball} \sin\left(\theta\right)^{3} - \mbox{Mcord} g \mbox{hrel} - 3 \, g \mbox{hrel} m\right)} \mbox{Wcord}} \sqrt{3} \cos\left(\theta\right)\right)}}{\mbox{Mcord} g + 3 \, g m}, t = \frac{\sqrt{\mbox{Mcord} + 3 \, m} \sqrt{-\mbox{dist} g \mbox{mball} \sin\left(\theta\right) + \mbox{Wcord}} \sqrt{2} \sqrt{3} \sin\left(\theta\right) - \sqrt{\mbox{Mcord} + 3 \, m} \sqrt{-3 \, \mbox{dist} g \mbox{mball} \sin\left(\theta\right)^{3} + \mbox{Mcord} g \mbox{hrel} + 3 \, \mbox{Wcord} \sin\left(\theta\right)^{2} + 3 \, g \mbox{hrel} m} \sqrt{2}}{\mbox{Mcord} g + 3 \, g m}\right], \left[\mbox{xf} = -\frac{2 \, {\left(3 \, \mbox{dist} g \mbox{mball} \sin\left(\theta\right)^{2} \cos\left(\theta\right) - 3 \, \mbox{Wcord} \sin\left(\theta\right) \cos\left(\theta\right) - \sqrt{3 \, \mbox{dist}^{2} g^{2} \mbox{mball}^{2} \sin\left(\theta\right)^{4} + 3 \, \mbox{Wcord}^{2} \sin\left(\theta\right)^{2} - {\left(\mbox{Mcord} \mbox{dist} g^{2} \mbox{hrel} \mbox{mball} + 3 \, \mbox{dist} g^{2} \mbox{hrel} m \mbox{mball}\right)} \sin\left(\theta\right) - {\left(6 \, \mbox{dist} g \mbox{mball} \sin\left(\theta\right)^{3} - \mbox{Mcord} g \mbox{hrel} - 3 \, g \mbox{hrel} m\right)} \mbox{Wcord}} \sqrt{3} \cos\left(\theta\right)\right)}}{\mbox{Mcord} g + 3 \, g m}, t = \frac{\sqrt{\mbox{Mcord} + 3 \, m} \sqrt{-\mbox{dist} g \mbox{mball} \sin\left(\theta\right) + \mbox{Wcord}} \sqrt{2} \sqrt{3} \sin\left(\theta\right) + \sqrt{\mbox{Mcord} + 3 \, m} \sqrt{-3 \, \mbox{dist} g \mbox{mball} \sin\left(\theta\right)^{3} + \mbox{Mcord} g \mbox{hrel} + 3 \, \mbox{Wcord} \sin\left(\theta\right)^{2} + 3 \, g \mbox{hrel} m} \sqrt{2}}{\mbox{Mcord} g + 3 \, g m}\right]\right]
 
 
 
 
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[\mbox{xf} = -\frac{2 \, {\left(3 \, \mbox{dist} g \mbox{mball} \sin\left(\theta\right)^{2} \cos\left(\theta\right) - 3 \, \mbox{Wcord} \sin\left(\theta\right) \cos\left(\theta\right) + \sqrt{3 \, \mbox{dist}^{2} g^{2} \mbox{mball}^{2} \sin\left(\theta\right)^{4} + 3 \, \mbox{Wcord}^{2} \sin\left(\theta\right)^{2} - {\left(\mbox{Mcord} \mbox{dist} g^{2} \mbox{hrel} \mbox{mball} + 3 \, \mbox{dist} g^{2} \mbox{hrel} m \mbox{mball}\right)} \sin\left(\theta\right) - {\left(6 \, \mbox{dist} g \mbox{mball} \sin\left(\theta\right)^{3} - \mbox{Mcord} g \mbox{hrel} - 3 \, g \mbox{hrel} m\right)} \mbox{Wcord}} \sqrt{3} \cos\left(\theta\right)\right)}}{\mbox{Mcord} g + 3 \, g m}, t = \frac{\sqrt{\mbox{Mcord} + 3 \, m} \sqrt{-\mbox{dist} g \mbox{mball} \sin\left(\theta\right) + \mbox{Wcord}} \sqrt{2} \sqrt{3} \sin\left(\theta\right) - \sqrt{\mbox{Mcord} + 3 \, m} \sqrt{-3 \, \mbox{dist} g \mbox{mball} \sin\left(\theta\right)^{3} + \mbox{Mcord} g \mbox{hrel} + 3 \, \mbox{Wcord} \sin\left(\theta\right)^{2} + 3 \, g \mbox{hrel} m} \sqrt{2}}{\mbox{Mcord} g + 3 \, g m}\right], \left[\mbox{xf} = -\frac{2 \, {\left(3 \, \mbox{dist} g \mbox{mball} \sin\left(\theta\right)^{2} \cos\left(\theta\right) - 3 \, \mbox{Wcord} \sin\left(\theta\right) \cos\left(\theta\right) - \sqrt{3 \, \mbox{dist}^{2} g^{2} \mbox{mball}^{2} \sin\left(\theta\right)^{4} + 3 \, \mbox{Wcord}^{2} \sin\left(\theta\right)^{2} - {\left(\mbox{Mcord} \mbox{dist} g^{2} \mbox{hrel} \mbox{mball} + 3 \, \mbox{dist} g^{2} \mbox{hrel} m \mbox{mball}\right)} \sin\left(\theta\right) - {\left(6 \, \mbox{dist} g \mbox{mball} \sin\left(\theta\right)^{3} - \mbox{Mcord} g \mbox{hrel} - 3 \, g \mbox{hrel} m\right)} \mbox{Wcord}} \sqrt{3} \cos\left(\theta\right)\right)}}{\mbox{Mcord} g + 3 \, g m}, t = \frac{\sqrt{\mbox{Mcord} + 3 \, m} \sqrt{-\mbox{dist} g \mbox{mball} \sin\left(\theta\right) + \mbox{Wcord}} \sqrt{2} \sqrt{3} \sin\left(\theta\right) + \sqrt{\mbox{Mcord} + 3 \, m} \sqrt{-3 \, \mbox{dist} g \mbox{mball} \sin\left(\theta\right)^{3} + \mbox{Mcord} g \mbox{hrel} + 3 \, \mbox{Wcord} \sin\left(\theta\right)^{2} + 3 \, g \mbox{hrel} m} \sqrt{2}}{\mbox{Mcord} g + 3 \, g m}\right]\right]
#kinematics, ugly print(s) 
        
[
[xf == -2*(3*dist*g*mball*sin(theta)^2*cos(theta) -
3*Wcord*sin(theta)*cos(theta) + sqrt(3*dist^2*g^2*mball^2*sin(theta)^4 +
3*Wcord^2*sin(theta)^2 - (Mcord*dist*g^2*hrel*mball +
3*dist*g^2*hrel*m*mball)*sin(theta) - (6*dist*g*mball*sin(theta)^3 -
Mcord*g*hrel - 3*g*hrel*m)*Wcord)*sqrt(3)*cos(theta))/(Mcord*g + 3*g*m),
t == (sqrt(Mcord + 3*m)*sqrt(-dist*g*mball*sin(theta) +
Wcord)*sqrt(2)*sqrt(3)*sin(theta) - sqrt(Mcord +
3*m)*sqrt(-3*dist*g*mball*sin(theta)^3 + Mcord*g*hrel +
3*Wcord*sin(theta)^2 + 3*g*hrel*m)*sqrt(2))/(Mcord*g + 3*g*m)],
[xf == -2*(3*dist*g*mball*sin(theta)^2*cos(theta) -
3*Wcord*sin(theta)*cos(theta) - sqrt(3*dist^2*g^2*mball^2*sin(theta)^4 +
3*Wcord^2*sin(theta)^2 - (Mcord*dist*g^2*hrel*mball +
3*dist*g^2*hrel*m*mball)*sin(theta) - (6*dist*g*mball*sin(theta)^3 -
Mcord*g*hrel - 3*g*hrel*m)*Wcord)*sqrt(3)*cos(theta))/(Mcord*g + 3*g*m),
t == (sqrt(Mcord + 3*m)*sqrt(-dist*g*mball*sin(theta) +
Wcord)*sqrt(2)*sqrt(3)*sin(theta) + sqrt(Mcord +
3*m)*sqrt(-3*dist*g*mball*sin(theta)^3 + Mcord*g*hrel +
3*Wcord*sin(theta)^2 + 3*g*hrel*m)*sqrt(2))/(Mcord*g + 3*g*m)]
]
[
[xf == -2*(3*dist*g*mball*sin(theta)^2*cos(theta) - 3*Wcord*sin(theta)*cos(theta) + sqrt(3*dist^2*g^2*mball^2*sin(theta)^4 + 3*Wcord^2*sin(theta)^2 - (Mcord*dist*g^2*hrel*mball + 3*dist*g^2*hrel*m*mball)*sin(theta) - (6*dist*g*mball*sin(theta)^3 - Mcord*g*hrel - 3*g*hrel*m)*Wcord)*sqrt(3)*cos(theta))/(Mcord*g + 3*g*m), t == (sqrt(Mcord + 3*m)*sqrt(-dist*g*mball*sin(theta) + Wcord)*sqrt(2)*sqrt(3)*sin(theta) - sqrt(Mcord + 3*m)*sqrt(-3*dist*g*mball*sin(theta)^3 + Mcord*g*hrel + 3*Wcord*sin(theta)^2 + 3*g*hrel*m)*sqrt(2))/(Mcord*g + 3*g*m)],
[xf == -2*(3*dist*g*mball*sin(theta)^2*cos(theta) - 3*Wcord*sin(theta)*cos(theta) - sqrt(3*dist^2*g^2*mball^2*sin(theta)^4 + 3*Wcord^2*sin(theta)^2 - (Mcord*dist*g^2*hrel*mball + 3*dist*g^2*hrel*m*mball)*sin(theta) - (6*dist*g*mball*sin(theta)^3 - Mcord*g*hrel - 3*g*hrel*m)*Wcord)*sqrt(3)*cos(theta))/(Mcord*g + 3*g*m), t == (sqrt(Mcord + 3*m)*sqrt(-dist*g*mball*sin(theta) + Wcord)*sqrt(2)*sqrt(3)*sin(theta) + sqrt(Mcord + 3*m)*sqrt(-3*dist*g*mball*sin(theta)^3 + Mcord*g*hrel + 3*Wcord*sin(theta)^2 + 3*g*hrel*m)*sqrt(2))/(Mcord*g + 3*g*m)]
]