Symbolic Catapult

376 days ago by roosh

t, theta, l, v0, g, x0, xf = var('t, theta, l, v0, g, x0, xf') eq1 = 0 == l* sin(theta) + v0*sin(theta) *t - .5* (g) * t^2 eq2 = xf == x0 + v0*cos(theta) *t solve([eq1,eq2],xf, t) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[\mbox{xf} = \frac{v_{0}^{2} \sin\left(\theta\right) \cos\left(\theta\right) - \sqrt{v_{0}^{2} \sin\left(\theta\right)^{2} + 2 \, g l \sin\left(\theta\right)} v_{0} \cos\left(\theta\right) + g x_{0}}{g}, t = \frac{v_{0} \sin\left(\theta\right) - \sqrt{v_{0}^{2} \sin\left(\theta\right) + 2 \, g l} \sqrt{\sin\left(\theta\right)}}{g}\right], \left[\mbox{xf} = \frac{v_{0}^{2} \sin\left(\theta\right) \cos\left(\theta\right) + \sqrt{v_{0}^{2} \sin\left(\theta\right)^{2} + 2 \, g l \sin\left(\theta\right)} v_{0} \cos\left(\theta\right) + g x_{0}}{g}, t = \frac{v_{0} \sin\left(\theta\right) + \sqrt{v_{0}^{2} \sin\left(\theta\right) + 2 \, g l} \sqrt{\sin\left(\theta\right)}}{g}\right]\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[\mbox{xf} = \frac{v_{0}^{2} \sin\left(\theta\right) \cos\left(\theta\right) - \sqrt{v_{0}^{2} \sin\left(\theta\right)^{2} + 2 \, g l \sin\left(\theta\right)} v_{0} \cos\left(\theta\right) + g x_{0}}{g}, t = \frac{v_{0} \sin\left(\theta\right) - \sqrt{v_{0}^{2} \sin\left(\theta\right) + 2 \, g l} \sqrt{\sin\left(\theta\right)}}{g}\right], \left[\mbox{xf} = \frac{v_{0}^{2} \sin\left(\theta\right) \cos\left(\theta\right) + \sqrt{v_{0}^{2} \sin\left(\theta\right)^{2} + 2 \, g l \sin\left(\theta\right)} v_{0} \cos\left(\theta\right) + g x_{0}}{g}, t = \frac{v_{0} \sin\left(\theta\right) + \sqrt{v_{0}^{2} \sin\left(\theta\right) + 2 \, g l} \sqrt{\sin\left(\theta\right)}}{g}\right]\right]
Wspring, m, t, theta, l, g, x0, xf = var('Wspring, m, t, theta, l, g, x0, xf') v0 = sqrt(2*(Wspring- m*g*l* sin(theta)) / m) eq1 = 0 == l* sin(theta) + v0*sin(theta) *t - .5* (g) * t^2 eq2 = xf == x0 + v0*cos(theta) *t solve([eq1,eq2],xf, t) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[\mbox{xf} = -\frac{2 \, g l m \sin\left(\theta\right)^{2} \cos\left(\theta\right) - 2 \, \mbox{Wspring} \sin\left(\theta\right) \cos\left(\theta\right) - g m x_{0} + 2 \, \sqrt{g^{2} l^{2} m^{2} \sin\left(\theta\right)^{4} - g^{2} l^{2} m^{2} \sin\left(\theta\right)^{2} + \mbox{Wspring}^{2} \sin\left(\theta\right)^{2} - {\left(2 \, g l m \sin\left(\theta\right)^{3} - g l m \sin\left(\theta\right)\right)} \mbox{Wspring}} \cos\left(\theta\right)}{g m}, t = \frac{\sqrt{-g l m \sin\left(\theta\right) + \mbox{Wspring}} \sqrt{2} \sqrt{m} \sin\left(\theta\right) - \sqrt{-g l m \sin\left(\theta\right)^{2} + g l m + \mbox{Wspring} \sin\left(\theta\right)} \sqrt{2} \sqrt{m} \sqrt{\sin\left(\theta\right)}}{g m}\right], \left[\mbox{xf} = -\frac{2 \, g l m \sin\left(\theta\right)^{2} \cos\left(\theta\right) - 2 \, \mbox{Wspring} \sin\left(\theta\right) \cos\left(\theta\right) - g m x_{0} - 2 \, \sqrt{g^{2} l^{2} m^{2} \sin\left(\theta\right)^{4} - g^{2} l^{2} m^{2} \sin\left(\theta\right)^{2} + \mbox{Wspring}^{2} \sin\left(\theta\right)^{2} - {\left(2 \, g l m \sin\left(\theta\right)^{3} - g l m \sin\left(\theta\right)\right)} \mbox{Wspring}} \cos\left(\theta\right)}{g m}, t = \frac{\sqrt{-g l m \sin\left(\theta\right) + \mbox{Wspring}} \sqrt{2} \sqrt{m} \sin\left(\theta\right) + \sqrt{-g l m \sin\left(\theta\right)^{2} + g l m + \mbox{Wspring} \sin\left(\theta\right)} \sqrt{2} \sqrt{m} \sqrt{\sin\left(\theta\right)}}{g m}\right]\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[\mbox{xf} = -\frac{2 \, g l m \sin\left(\theta\right)^{2} \cos\left(\theta\right) - 2 \, \mbox{Wspring} \sin\left(\theta\right) \cos\left(\theta\right) - g m x_{0} + 2 \, \sqrt{g^{2} l^{2} m^{2} \sin\left(\theta\right)^{4} - g^{2} l^{2} m^{2} \sin\left(\theta\right)^{2} + \mbox{Wspring}^{2} \sin\left(\theta\right)^{2} - {\left(2 \, g l m \sin\left(\theta\right)^{3} - g l m \sin\left(\theta\right)\right)} \mbox{Wspring}} \cos\left(\theta\right)}{g m}, t = \frac{\sqrt{-g l m \sin\left(\theta\right) + \mbox{Wspring}} \sqrt{2} \sqrt{m} \sin\left(\theta\right) - \sqrt{-g l m \sin\left(\theta\right)^{2} + g l m + \mbox{Wspring} \sin\left(\theta\right)} \sqrt{2} \sqrt{m} \sqrt{\sin\left(\theta\right)}}{g m}\right], \left[\mbox{xf} = -\frac{2 \, g l m \sin\left(\theta\right)^{2} \cos\left(\theta\right) - 2 \, \mbox{Wspring} \sin\left(\theta\right) \cos\left(\theta\right) - g m x_{0} - 2 \, \sqrt{g^{2} l^{2} m^{2} \sin\left(\theta\right)^{4} - g^{2} l^{2} m^{2} \sin\left(\theta\right)^{2} + \mbox{Wspring}^{2} \sin\left(\theta\right)^{2} - {\left(2 \, g l m \sin\left(\theta\right)^{3} - g l m \sin\left(\theta\right)\right)} \mbox{Wspring}} \cos\left(\theta\right)}{g m}, t = \frac{\sqrt{-g l m \sin\left(\theta\right) + \mbox{Wspring}} \sqrt{2} \sqrt{m} \sin\left(\theta\right) + \sqrt{-g l m \sin\left(\theta\right)^{2} + g l m + \mbox{Wspring} \sin\left(\theta\right)} \sqrt{2} \sqrt{m} \sqrt{\sin\left(\theta\right)}}{g m}\right]\right]
v0 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\sqrt{-\frac{g l m \sin\left(\theta\right) - \mbox{Wspring}}{m}} \sqrt{2}
\newcommand{\Bold}[1]{\mathbf{#1}}\sqrt{-\frac{g l m \sin\left(\theta\right) - \mbox{Wspring}}{m}} \sqrt{2}
4/5 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{4}{5}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{4}{5}
 
       
 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[\mbox{xf} = -\frac{2 \, g l m \sin\left(\theta\right)^{2} \cos\left(\theta\right) - 2 \, \mbox{Wspring} \sin\left(\theta\right) \cos\left(\theta\right) - g m x_{0} + 2 \, \sqrt{g^{2} l^{2} m^{2} \sin\left(\theta\right)^{4} - g^{2} l^{2} m^{2} \sin\left(\theta\right)^{2} + \mbox{Wspring}^{2} \sin\left(\theta\right)^{2} - {\left(2 \, g l m \sin\left(\theta\right)^{3} - g l m \sin\left(\theta\right)\right)} \mbox{Wspring}} \cos\left(\theta\right)}{g m}, t = \frac{\sqrt{-g l m \sin\left(\theta\right) + \mbox{Wspring}} \sqrt{2} \sqrt{m} \sin\left(\theta\right) - \sqrt{-g l m \sin\left(\theta\right)^{2} + g l m + \mbox{Wspring} \sin\left(\theta\right)} \sqrt{2} \sqrt{m} \sqrt{\sin\left(\theta\right)}}{g m}\right], \left[\mbox{xf} = -\frac{2 \, g l m \sin\left(\theta\right)^{2} \cos\left(\theta\right) - 2 \, \mbox{Wspring} \sin\left(\theta\right) \cos\left(\theta\right) - g m x_{0} - 2 \, \sqrt{g^{2} l^{2} m^{2} \sin\left(\theta\right)^{4} - g^{2} l^{2} m^{2} \sin\left(\theta\right)^{2} + \mbox{Wspring}^{2} \sin\left(\theta\right)^{2} - {\left(2 \, g l m \sin\left(\theta\right)^{3} - g l m \sin\left(\theta\right)\right)} \mbox{Wspring}} \cos\left(\theta\right)}{g m}, t = \frac{\sqrt{-g l m \sin\left(\theta\right) + \mbox{Wspring}} \sqrt{2} \sqrt{m} \sin\left(\theta\right) + \sqrt{-g l m \sin\left(\theta\right)^{2} + g l m + \mbox{Wspring} \sin\left(\theta\right)} \sqrt{2} \sqrt{m} \sqrt{\sin\left(\theta\right)}}{g m}\right]\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[\mbox{xf} = -\frac{2 \, g l m \sin\left(\theta\right)^{2} \cos\left(\theta\right) - 2 \, \mbox{Wspring} \sin\left(\theta\right) \cos\left(\theta\right) - g m x_{0} + 2 \, \sqrt{g^{2} l^{2} m^{2} \sin\left(\theta\right)^{4} - g^{2} l^{2} m^{2} \sin\left(\theta\right)^{2} + \mbox{Wspring}^{2} \sin\left(\theta\right)^{2} - {\left(2 \, g l m \sin\left(\theta\right)^{3} - g l m \sin\left(\theta\right)\right)} \mbox{Wspring}} \cos\left(\theta\right)}{g m}, t = \frac{\sqrt{-g l m \sin\left(\theta\right) + \mbox{Wspring}} \sqrt{2} \sqrt{m} \sin\left(\theta\right) - \sqrt{-g l m \sin\left(\theta\right)^{2} + g l m + \mbox{Wspring} \sin\left(\theta\right)} \sqrt{2} \sqrt{m} \sqrt{\sin\left(\theta\right)}}{g m}\right], \left[\mbox{xf} = -\frac{2 \, g l m \sin\left(\theta\right)^{2} \cos\left(\theta\right) - 2 \, \mbox{Wspring} \sin\left(\theta\right) \cos\left(\theta\right) - g m x_{0} - 2 \, \sqrt{g^{2} l^{2} m^{2} \sin\left(\theta\right)^{4} - g^{2} l^{2} m^{2} \sin\left(\theta\right)^{2} + \mbox{Wspring}^{2} \sin\left(\theta\right)^{2} - {\left(2 \, g l m \sin\left(\theta\right)^{3} - g l m \sin\left(\theta\right)\right)} \mbox{Wspring}} \cos\left(\theta\right)}{g m}, t = \frac{\sqrt{-g l m \sin\left(\theta\right) + \mbox{Wspring}} \sqrt{2} \sqrt{m} \sin\left(\theta\right) + \sqrt{-g l m \sin\left(\theta\right)^{2} + g l m + \mbox{Wspring} \sin\left(\theta\right)} \sqrt{2} \sqrt{m} \sqrt{\sin\left(\theta\right)}}{g m}\right]\right]
 
       
Wspring, m, t, theta, l, g, x0, xf = var('Wspring, m, t, theta, l, g, x0, xf') v0 = sqrt(2*(Wspring- m*g*l* sin(theta)) / m) eq1 = 0 == l* sin(theta) + v0*sin(theta) *t - .5* (g) * t^2 eq2 = xf == x0 + v0*cos(theta) *t solve([eq1,eq2],xf, t) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[\mbox{xf} = -\frac{2 \, g l m \sin\left(\theta\right)^{2} \cos\left(\theta\right) - 2 \, \mbox{Wspring} \sin\left(\theta\right) \cos\left(\theta\right) - g m x_{0} + 2 \, \sqrt{g^{2} l^{2} m^{2} \sin\left(\theta\right)^{4} - g^{2} l^{2} m^{2} \sin\left(\theta\right)^{2} + \mbox{Wspring}^{2} \sin\left(\theta\right)^{2} - {\left(2 \, g l m \sin\left(\theta\right)^{3} - g l m \sin\left(\theta\right)\right)} \mbox{Wspring}} \cos\left(\theta\right)}{g m}, t = \frac{\sqrt{-g l m \sin\left(\theta\right) + \mbox{Wspring}} \sqrt{2} \sqrt{m} \sin\left(\theta\right) - \sqrt{-g l m \sin\left(\theta\right)^{2} + g l m + \mbox{Wspring} \sin\left(\theta\right)} \sqrt{2} \sqrt{m} \sqrt{\sin\left(\theta\right)}}{g m}\right], \left[\mbox{xf} = -\frac{2 \, g l m \sin\left(\theta\right)^{2} \cos\left(\theta\right) - 2 \, \mbox{Wspring} \sin\left(\theta\right) \cos\left(\theta\right) - g m x_{0} - 2 \, \sqrt{g^{2} l^{2} m^{2} \sin\left(\theta\right)^{4} - g^{2} l^{2} m^{2} \sin\left(\theta\right)^{2} + \mbox{Wspring}^{2} \sin\left(\theta\right)^{2} - {\left(2 \, g l m \sin\left(\theta\right)^{3} - g l m \sin\left(\theta\right)\right)} \mbox{Wspring}} \cos\left(\theta\right)}{g m}, t = \frac{\sqrt{-g l m \sin\left(\theta\right) + \mbox{Wspring}} \sqrt{2} \sqrt{m} \sin\left(\theta\right) + \sqrt{-g l m \sin\left(\theta\right)^{2} + g l m + \mbox{Wspring} \sin\left(\theta\right)} \sqrt{2} \sqrt{m} \sqrt{\sin\left(\theta\right)}}{g m}\right]\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[\left[\mbox{xf} = -\frac{2 \, g l m \sin\left(\theta\right)^{2} \cos\left(\theta\right) - 2 \, \mbox{Wspring} \sin\left(\theta\right) \cos\left(\theta\right) - g m x_{0} + 2 \, \sqrt{g^{2} l^{2} m^{2} \sin\left(\theta\right)^{4} - g^{2} l^{2} m^{2} \sin\left(\theta\right)^{2} + \mbox{Wspring}^{2} \sin\left(\theta\right)^{2} - {\left(2 \, g l m \sin\left(\theta\right)^{3} - g l m \sin\left(\theta\right)\right)} \mbox{Wspring}} \cos\left(\theta\right)}{g m}, t = \frac{\sqrt{-g l m \sin\left(\theta\right) + \mbox{Wspring}} \sqrt{2} \sqrt{m} \sin\left(\theta\right) - \sqrt{-g l m \sin\left(\theta\right)^{2} + g l m + \mbox{Wspring} \sin\left(\theta\right)} \sqrt{2} \sqrt{m} \sqrt{\sin\left(\theta\right)}}{g m}\right], \left[\mbox{xf} = -\frac{2 \, g l m \sin\left(\theta\right)^{2} \cos\left(\theta\right) - 2 \, \mbox{Wspring} \sin\left(\theta\right) \cos\left(\theta\right) - g m x_{0} - 2 \, \sqrt{g^{2} l^{2} m^{2} \sin\left(\theta\right)^{4} - g^{2} l^{2} m^{2} \sin\left(\theta\right)^{2} + \mbox{Wspring}^{2} \sin\left(\theta\right)^{2} - {\left(2 \, g l m \sin\left(\theta\right)^{3} - g l m \sin\left(\theta\right)\right)} \mbox{Wspring}} \cos\left(\theta\right)}{g m}, t = \frac{\sqrt{-g l m \sin\left(\theta\right) + \mbox{Wspring}} \sqrt{2} \sqrt{m} \sin\left(\theta\right) + \sqrt{-g l m \sin\left(\theta\right)^{2} + g l m + \mbox{Wspring} \sin\left(\theta\right)} \sqrt{2} \sqrt{m} \sqrt{\sin\left(\theta\right)}}{g m}\right]\right]