Motion In A Plane

c) Position vector of the particle
               \( \vec U = \left( {u\cos \theta } \right)\hat{i} + \left( {u\sin \theta } \right)\hat{j} \) 
                                     
        
          \( \overrightarrow a = - g\hat{j} \)        
           \( \overrightarrow s = \overrightarrow u t + \frac{1} {2}\overrightarrow a t^2 \)
          \( \overrightarrow r = \left[ {\left( {u\cos \theta } \right)\hat{i} + \left( {u\sin \theta } \right)\hat{j}} \right]t + \frac{1} {2}\left( { - g\hat j} \right)t^2 \)
          \( \overrightarrow r = (u\cos \theta \,t)\hat{i} + \left( {u\sin \theta \,t - \frac{1} {2}gt^2 } \right)\hat j \)
           \( \overrightarrow r = \text{x}\,\,\hat{i} + y\hat j \)     \( x = (u\cos \theta )t \)  
          \( y = (u\sin \theta )t - \frac{1} {2}gt^2 \)
    Magnitude of displacement of projecticle    S = \( \sqrt {x^2 + y^2 } \)     
    angle made by the position vector with horizontal.    
      \( Tan\,\alpha = \frac{{(U\sin \theta )t - \frac{1} {2}gt^2 }} {{(U\cos \theta )t}} \)
            Tan \( \alpha = \frac{{U\sin \theta - \frac{1} {2}gt}} {{U\cos \theta }} \)