Three particles are projected in the air with the minimum possible speeds, such that the first goes from A to B , the second goes from B to C and the third goes from C to A . Points A and C are at the same horizontal level. The two inclines make the same angle α with the horizontal, as shown. The relation among the projection speeds of the three particles is

Step-by-step solution

For a given speed, the maximum range of a projectile on a horizontal level is R max = u 2 g the maximum range up and down the incline are R up max = u 2 g 1 + sin α R down max = u 2 g 1 - sin α by substituting the values of the velocities given in the problem, we get R = u 2 2 g 1 + sin ⁡ α ⇒ u 2 2 = R g 1 + sin ⁡ α     … ( 1 ) 2 R cos ⁡ α =   u 3 2 g ⇒ u 3 2   = 2 R g cos ⁡ α     … ( 2 ) Now, u 1 u 2 = R g 1