Friction is a very clever force

IMG 20170219 105822.jpg

The particle is incident as shown in the figure .The impulse provided by friction and normal reaction are as shown in the figure .

From the coefficient of restitution equation,

$\displaystyle \frac{0 - v\cos{\theta} }{-u\cos{\theta}} = e$

$\displaystyle \Rightarrow \color{#3D99F6}{ v = eu}$

Hence, Impulse provided by Normal reaction force is

$\displaystyle K= m ( v\cos{\theta} - (-u\cos{\theta})) = m(1+ e)u\cos{\theta}$

Impulse by friction is

$\displaystyle J= \mu K = m\mu (1+e)u\cos{\theta}$

$\displaystyle -m(v\sin{\theta} - u\sin{\theta} ) = m\mu (1+ e)u\cos{\theta}$

$\displaystyle mu\sin{\theta}(1-e ) = mu~ \mu \cos{\theta}(1+e). ~~~~~~~~~~~~~~~~~~~\cdots (using ~\color{#D61F06}{ v = eu })$

$\displaystyle \Rightarrow \mu = \frac{(1-e)\tan{\theta}}{1+e} = \frac{1}{2} = \frac{5}{10}$

$\implies \color{#3D99F6}{X= 5}$

$Answer = 5 + 10 + 15 = \color{#3D99F6}{30}$

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