A block A of mass 1 kg is projected along rough horizontal surface from $\mathrm{x}=0$ with speed ${\mathrm{v}}_0=\sqrt{10}$ m/s. Find the distance travelled by block before coming to rest. The coefficient of kinetic friction between block and the surface is $\text{μ}=\mathrm{bx},$$\mathrm{b}=1{\text{\hspace{0.17em}\hspace{0.17em}m}}^{-1}.$  where   ( $\mathrm{g}=10\hspace{0.17em}\hspace{0.17em}\mathrm{m}/{\mathrm{s}}^2$)

Block Projected on Rough Horizontal Surface

Given:

Mass of block, m = 1 kg

Initial speed, v₀ = √10 m/s 
Coefficient of kinetic friction, μ = b x, b = 1 m^-1
Acceleration due to gravity, g = 10 m/s²

Solution Steps:

1. Kinetic energy lost = Work done by friction
   Work done by friction:
   F_fric = μ N = b x m g 
   W = ∫₀^x bx m g dx = m g b [x²/2]₀^x = (1/2) m g b x²
2. Equate work done to initial kinetic energy: 
   (1/2) m v₀² = (1/2) m g b x²
3. Simplifying: v₀² = g b x²
   x = √(v₀² / g b)
4. Substitute values:
   x = √(( √10 )² / (10 × 1)) 
   x = √(10/10) = √1 = 1 m

Final Answer: The block travels 1 meter before coming to rest.