Motion In A Plane

§§ Vector notations incase of projectile motion

a) Velocity after some time : Initital velocity the projectile 
        \( \overrightarrow v = \left( {u\cos \theta } \right)\hat{i} + \left( {u\sin \theta } \right)\hat{j} \)                  
              \( \overrightarrow a = - g\hat{j} \)                                                  
    Velocity of the projectile after some time \( \overrightarrow v = \overrightarrow u + \overrightarrow a t \)                                                             
     \( \overrightarrow v = \left( {u\cos \theta } \right)\hat{i }+ \left( {u\sin \theta - gt} \right)\hat{j} \) = \( v_x \hat{i} + v_y \hat{j} \)  
         

    \( v = \sqrt {\left( {u\cos \theta } \right)^2 + \left( {u\sin \theta - gt} \right)^2 } \)
 angle made by the velocity with horizontal. \( Tan\alpha = \frac{{v_y }} {{v_x }} = \frac{{U\sin \theta - gt}} {{U\cos \theta }} \)