A projectile is fired on a horizontal ground. Coefficient of restitution between the projectile and the ground is e. If a, b and c be the ratio of time of flight $[\frac{T_{1}}{T_{2}}]$ maximum height $[\frac{H_{1}}{H_{2}}]$ and horizontal range $[\frac{R_{1}}{R_{2}}]$ in first two collisions with the ground. Then

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a = \frac{1}{e}

b = \frac{1}{e^{2}}

c = \frac{1}{e}

All the above

answer is D.

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Detailed Solution

$\frac{T_{1}}{T_{2}} = a$

$\frac{H_{1}}{H_{2}} = b$

$\frac{R_{1}}{R_{2}} = c$

$T = \frac{2 u_{x} u_{y}}{g}$

$T \propto u_{y}$

$\frac{T_{1}}{T_{2}} = \frac{{(u_{1})}_{y}}{{(u_{2})}_{y}} = \frac{u_{1} y}{e (u_{1} y)} = \frac{1}{e}$

$∴ a = \frac{1}{e}$

$H = \frac{u^{2} g}{2g} \Rightarrow H \propto u_{y}^{2}$

$∴ \frac{H_{1}}{H_{2}} = \frac{{(u_{1})}_{y}^{2}}{{(u_{2})}_{y}^{2}} = \frac{{(u_{1})}_{y}^{2}}{e^{2} {(u_{1})}_{y}^{2}} = \frac{1}{e^{2}}$

$∴ b = \frac{1}{e^{2}}$

$R = u_{x} . T$

$R \propto T$

$but \frac{T_{1}}{T_{2}} = \frac{1}{e}$

$\Rightarrow \frac{R_{1}}{R_{2}} = c = \frac{1}{e}$

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