A block of mass m is projected with an initial velocity v₀ on a rough horizontal surface. The coefficient of kinetic friction between the block and the surface increases linearly with the distance x from the starting point as _k = bx , where b is a positive constant. The distance D at which the block comes to rest is :

Step-by-step solution

According to the Work-Energy Theorem, the net work done on the block equals its change in kinetic energy: W_ net = K The frictional force acting on the block at a distance x is f_k = _k mg = bxmg . The work done by friction as the block moves from x=0 to x=D is: W_f = ₀^ D (-f_k) dx = - ₀^ D bxmg dx = -bmg [ x^2 2 ]₀^ D = - 1 2 bmgD^2 The change in kinetic energy is K = K_f - K_i = 0 - 1 2 mv₀^2 Equating the two: - 1 2 bmgD^2 = - 1 2 mv₀^2 bmgD^2 = mv₀^2 D^2 = v₀^2 bg D = v₀ bg Answer: v₀ bg