Q: A block A of mass 1 kg is projected along rough horizontal surface from x = 0 with speed v0 = √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