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

Question

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, u 3 = u 1 + u 2, u 3 2 = 2 u 1 u 2, 1 u 3 = 1 u 1 + 1 u 2, u 3 2 = u 1 2 + u 2 2

MARKS App · Accepted Answer

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 - sin 2 ⁡ α = R g c o s α = u 3 2 2 ∴ u 3 2 = 2 u 1 u 2

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