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Newton's Second Law: Rotatory Version

Two particles...

Question

Two particles A and B are projected from the same point in different directions in such a manner that vertical components of their initial velocities are same in above figure. Find the the ratio of range.

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Solution

We know that the range of projectile,

$R = \frac{u^{2} sin 2 θ}{g} = \frac{(2 u sin θ)}{g} \times u cos θ = \frac{2 u_{x} u_{y}}{g}$

As, $(u_{y})_{1} = (u_{y})_{2} \Rightarrow u_{A} sin θ_{A} = u_{B} sin θ_{B}$

or $\frac{u_{A}}{u_{B}} = \frac{sin θ_{B}}{sin θ_{A}}$

$\frac{R_{A}}{R_{B}} = \frac{2 (u_{A} sin θ_{A}) (u_{A} cos θ_{A}) / g}{2 (u_{B} s i n θ_{B}) (u_{B} c o s θ_{B}) / g}$

$= [\frac{u_{A} cos θ_{A}}{u_{B} cos θ_{B}}]$

$= \frac{sin θ_{B} cos θ_{A}}{sin θ_{A} cos θ_{B}} = \frac{tan θ_{B}}{tan θ_{A}}$

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Newton's Second Law: Rotatory Version

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