Motion In A Plane

b) Velocity after some hegiht
        \( \begin{gathered} \overrightarrow v = u\cos \theta \,\hat{i} + \left( {u\sin \theta } \right)\hat{j} \hfill \\ \overrightarrow a = - g\hat j \hfill \\ \overrightarrow s = h\hat j \hfill \\ \overrightarrow v .\overrightarrow v = \overrightarrow u .\overrightarrow u + 2\overrightarrow a .\overrightarrow s \hfill \\ \end{gathered} \)
                             and    
          \( v^2 = \left( {ucos\theta } \right)^2 + \left( {u\sin \theta } \right)^2 - 2gh \) 
      
      
         \( v = \sqrt {\left( {ucos\theta } \right)^2 + \left[ {\left( {u\sin \theta } \right)^2 - 2gh} \right]} \)
In vector form we can write it as  \( \overrightarrow v = \left( {u\cos \theta } \right)\hat{i} + \left( {\sqrt {\left( {u\sin \theta } \right)^2 - 2gh} } \right)\hat{j} = v_x \hat i + v_y \hat j \)     
angle made by the velocity with horizontal.    
    \( Tan\alpha = \frac{{v_y }} {{v_x }} = \frac{{\sqrt {(U\sin \theta )^2 - 2gh} }} {{U\cos \theta }} \)