Q: A block A of mass 1 kg is projected along rough horizontal surface from x = 0 with speed v_{0} = √10m/s. Find the distance travelled by block before coming to rest. The coefficient of kinetic friction between block and the surface is µ = *bx*, where b = 1 m^{-1}.

(a) 1 m

(b) 2 m

(c) 3 m

(d) 4 m

Ans: (a)

Solution: Retardation of the block is

$ \displaystyle a = \frac{f}{m}= \frac{\mu m g}{m} $

$ \displaystyle a = \mu g = b x g $

$ \displaystyle -\frac{v dv}{dx} = x g $ (As b = 1 m^{-1} )

on integrating ,

$ \displaystyle -\int_{v_0}^{0} v dv = g \int_{0}^{x}x dx $

$ \displaystyle \frac{v_0 ^2}{2} = \frac{g x^2}{2} $

$ \displaystyle v_0 = \sqrt{g} x $

$ \displaystyle x = \frac{v_0}{\sqrt{g}} = \frac{\sqrt{10}}{\sqrt{10}} $

x = 1 m