Motion In A Plane

(d) Position, Time and speed at any angular elevation 
Let a particle be projected with v₀ at an angle \(\theta_0\) .After a time t it  moves with a velocity v at an angle \(\theta\) with horizontal as shown in the figure. Since the horizontal component of velocity of a projectile remains constant i.e.,
    
       

       \( v_x = v_0 \cos \theta _0 = v\cos \theta \) \( \Rightarrow v = \frac{{v_0 \cos \theta _0 }} {{\cos \theta }} \)    
As we know that v² = v₀² – 2gy, substituting the obtained value of V in this equation we obtain
            \( y = \frac{{v_0^2 - v^2 }} {{2g}} \) \( y = \frac{1} {{2g}}\left[ {v_0 ^2 - \left( {\frac{{v_0 \cos \theta _0 }} {{\cos \theta }}} \right)^2 } \right] \) \( \Rightarrow \,\,\,\,\,y = \frac{{v_0 ^2 }} {{2g}}\left( {1 - \frac{{\cos ^2 \theta _0 }} {{\cos ^2 \theta }}} \right) \)  
    We know that  v_y = (v_y)0 – gt \( \Rightarrow \,\,\,\,\,v\sin \theta = v_0 \sin \theta _0 - gt \)
    substituting \( v = \frac{{v_0 \cos \theta _0 }} {{\cos \theta }} \)  we obtain \( \left( {\frac{{v_0 \cos \theta _0 }} {{\cos \theta }}} \right)\sin \theta = v_0 \sin \theta _0 - gt \)   
        \( \Rightarrow t = \frac{{v_0 \sin \left( {\theta _0 - \theta } \right)}} {{g\cos \theta }} \)
The horizontal distance x covered during the time t is given as 
         \( x = (v_0 \cos \theta _0 )t \) \( \Rightarrow x = \left( {v_0 \cos \theta _0 } \right)\frac{{v_0 \sin \left( {\theta _0 - \theta } \right)}} {{g\cos \theta }} \) \( \Rightarrow x = \frac{{v_0^2 \sin \left( {\theta _0 - \theta } \right)\cos \theta _0 }} {{g\cos \theta }} \)  
                       

Referring to the adjoining figure when the velocity vector \(\vec v\) becomes perpendicular to the initial velocity vector \(\vec v_0\)
    \( v.\overrightarrow {v_0 } = 0, \) where \( v_0 = v_0 \cos \theta _0 \hat{i} + v_0 \sin \theta _0 \hat{j} \)
    and \( v = \left( {v_0 \cos \theta _0 } \right)\hat{i} + \left( {gt - v_0 \sin \theta _0 } \right)\left( { - \hat{j}} \right) \)  
     \( \Rightarrow \left( {\left( {v_0 \cos \theta _0 } \right)\hat{i} + \left( {v_0 \sin \theta _0 - gt} \right)\hat{j}} \right). \) \( \left( {v_0 \cos \theta _0 } \right)^2 + \left( {v_0 \sin \theta _0 - gt} \right)v_0 \sin \theta _0 = 0 \) 
\( \Rightarrow \,\,v_0 \left( {\sin ^2 \theta _0 + \cos ^2 \theta _0 } \right)\, = gt\sin \theta _0 \) \( \Rightarrow t = \frac{{v_0 }} {{g\sin \theta _{0`} }} \)