Oblique Projectile From Inclined Surface

iii) Maximum height (H_max):
The vertical displacement of a projectile during time of ascent is the maximum height of the projectile.
In the equation \( y = (u\sin \theta )t - \frac{1} {2}gt^2 \) we use y=H_max  and t=t_a
        \( \therefore H_{\max } = \frac{{u^2 \sin ^2 \theta }} {{2g}} = \frac{{u_y ^2 }} {{2g}} \)

Note : When  \(\theta\)= 90°,   H_max= \( \frac{{u^2 }} {{2g}} \)
This is equal to the maximum height reached by a body projected vertically upwards

iv) Horizontal Range (R) :
This is defined as the horizontal distance covered by projectile during its time of flight.
Thus, by definition,
    Range R= horizontal velocity X time of flight
    i.e., R=(ucos\(\theta\))T=(ucos\(\theta\))\( \frac{{2u\sin \theta }} {g} \)
      \( \Rightarrow Range,R = \frac{{u^2 \sin 2\theta }} {g} = \frac{{2u_x u_y }} {g} \)
                  

Note : For a given value of projection velocity u, R is maximum when 2\(\theta\)=90° ie., \(\theta\)=45°.
So maximum horizontal range is
                   R_max = \( \frac{{u^2 }} {g} \)(sin90° =1)

Note : For a given speed of projection 'u', the ranges are equal for angles
(a) \(\theta\) and (90- \(\theta\))
(b) (45+\(\alpha\)) and (45-\(\alpha\))