Question

What relation exist between the range covered by two projectiles thrown up with the same initial velocities at angles of projection that are complementary

Answer 1

Let 1 represent the first body and 2 represent the second body

$u_1 = u_2 = u$

$\theta_1 + \theta_2 = 90°$ (complementary angles)

$\therefore \theta_2 = 90° - \theta_1$

For body 1,

$R_1 = \dfrac{{u_1}^2\, sin2\theta_1}{g}$

$R_1 = \dfrac{u^2\, sin2\theta_1}{g}$

For body 2;

$R_2 = \dfrac{{u_2}^2\, sin2\theta_2}{g}$

$R_2 = \dfrac{{u}^2\, sin2(90-\theta_1)}{g}$

$R_2 = \dfrac{{u}^2\, sin(180-2\theta_1)}{g}$

$sin(180-\theta) = sin \theta\\ \textsf{Then, }sin(180 - 2\theta) = sin2\theta$

$\therefore R_2 = \dfrac{{u}^2\, sin2\theta_1}{g}$

$R_1 = R_2$

$\therefore$ The range covered by two projectiles thrown up with the same initial velocities at angles of projection that are complementary is equal