Question

What is the relationship between time of flight $T$ and horizontal range $R$ ?
(where $\theta$ is angle of projection with the horizontal)

• A. $R=\frac{gT}{tan\theta }$
• B. $R=\frac{g{T}^2}{2tan\theta }$
• C. $R=\frac{g{T}^2}{tan\theta }$
• D. $R=\frac{gT}{2tan\theta }$

Answer 1

Answer: (B)

Horizontal range $R=\frac{u^{2}sin2\theta }{g}$
Time of flight $T=\frac{2usin\theta }{g}$
where u is the velocity of the projection and $\theta$ is the angle of projection with the horizontal
$R=\frac{u^{2}sin2\theta }{g}$ or $R=\frac{2u^{2}sin\theta cos\theta }{g}$
or $\frac{R}{2cos\theta }=\frac{u^{2}sin\theta }{g}\dots \left(i\right)$
$T=\frac{2usin\theta }{g}$ or $T_2=\frac{4u^{2}si{n}^2\theta }{g^2}$
$T^2=\frac{4}{g}\left(\frac{u^{2}sin\theta }{g}\right)sin\theta$
$=\frac{4}{g}\left(\frac{R}{2cos\theta }\right)sin\theta$ (Using (i))
or $T_2=\frac{2R}{g}tan\theta$
or $R=\frac{g{T}^2}{2tan\theta }$