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

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

TestHub · Accepted Answer

For a given speed,

the maximum range of a projectile on a horizontal level is

$R_{\max} = \frac{u^{2}}{g}$

the maximum range up and down the incline are

${(R_{up})}_{\max} = \frac{u^{2}}{g (1 + \sin α)} {(R_{down})}_{\max} = \frac{u^{2}}{g (1 - \sin α)}$

by substituting the values of the velocities given in the problem, we get

$R = \frac{u_{2}^{2}}{g (1 + \sin α)} \Rightarrow u_{2}^{2} = R g (1 + \sin α) \dots (1)$
$2 R \cos α = \frac{u_{3}^{2}}{g} \Rightarrow u_{3}^{2} = 2 R g \cos α \dots (2)$
Now, $u_{1} u_{2} = R g \sqrt{1 - \sin^{2} α} = R g c o s α = \frac{u_{3}^{2}}{2}$
$∴ u_{3}^{2} = 2 u_{1} u_{2}$

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