Motion in a Plane

a) Time of ascent :-
Time Taken to Reach Maximum Height. We let this time be denoted t_a. At the maximum height the vertical velocity is zero i.e.,V_y=0  
                                      \( u\sin \theta - gt_a = 0 \)      \( \therefore t_a = \frac{{u\sin \theta }} {g} \)

b) Time of Flight : -
The total time spent by a projectile in air during the time of motion is called time of flight.
    On substituting y = 0 and t = T in equation (4.3) we can write 
       \( 0 = u\sin \theta T - \frac{1} {2}gT^2 \)     \( \Rightarrow T = \frac{{2u\sin \theta }} {g}\,\,..........(4.4) \)  

c) Maximum Height : -
As the projectile ascends (moves up), the vertical component of its velocity decreases. At maximum height of the vertical component of velocity becomes zero.
 That means the projectile moves horizontally at its highest position.
    From the equatrion of motion \( v^2 - u^2 = 2as \), the maximum height attained by a projectile can be written as
            \( (0)^2 - (u\sin \theta )^2 = 2 \times - g \times H_m \)                   \( \Rightarrow - u^2 \sin ^2 \theta = - 2gH_m \)
                        \( \Rightarrow H_m = \frac{{u^2 \sin ^2 \theta }} {{2g}}\,\,\,................\,(4.5) \)
Note: When.\( \theta = 90^0 ,H_{\max } = \frac{{u^2 }} {{2g}} \) This is equal to the maximum height reached by a body projected vertically upwards.