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(A) Step $1$: Calculate the velocity of the block at the bottom of the incline using the work-energy theorem or kinematics.Using $v^2 = u^2 + 2as$,whe

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(A) Step $1$: Calculate the velocity of the block at the bottom of the incline using the work-energy theorem or kinematics.Using $v^2 = u^2 + 2as$,where $u = 0$,$a = g \sin 30^\circ = 10 	imes 0.5 = 5 \, m/s^2$,and $s = 2 \, m$:$v^2 = 0 + 2 	imes 5 	imes 2 = 20 \, m^2/s^2$.Step $2$: Calculate the distance traveled on the rough surface.On the horizontal surface,the deceleration $a'$ is caused by friction: $a' = \mu g = 0.25 	imes 10 = 2.5 \, m/s^2$.Using $v_f^2 = v^2 - 2a'S$,where $v_f = 0$ (final velocity is zero):$0 = 20 - 2 	imes 2.5 	imes S$.$5S = 20 \implies S = 4 \, m$.

$A$ $2 \, kg$ block is placed on a frictionless inclined plane with an angle of $30^\circ$ and a length of $2 \, m$. After reaching the bottom of the incline,it travels on a surface with a coefficient of friction of $0.25$. What distance (in $m$) will it travel on the surface before coming to rest?

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