Two projectiles A and B are launched from the ground at angles of 30^ and 60^ with the horizontal, respectively. If both projectiles attain the same maximum height, the ratio of their horizontal ranges R_A : R_B is

Step-by-step solution

The maximum height of a projectile is given by H = u^2 ^2 2g . Given that H_A = H_B : u_A^2 ^2 30^ 2g = u_B^2 ^2 60^ 2g u_A^2 ( 1 2 )^2 = u_B^2 ( 3 2 )^2 u_A^2 4 = 3u_B^2 4 u_A^2 u_B^2 = 3 The horizontal range is given by R = u^2 2 g . The ratio of their ranges is: R_A R_B = u_A^2 (2 30^ ) g u_B^2 (2 60^ ) g = ( u_A^2 u_B^2 ) 60^ 120^ Since 60^ = 120^ = 3 2 , the ratio simplifies to: R_A R_B = u_A^2 u_B^2 = 3 Thus, the ratio is 3 : 1 . Answer: 3 : 1