A particle of mass m moving along the positive x -axis experiences a position-dependent retarding force F = -k x , where k is a positive constant. If the particle enters the region at x=0 with a velocity v₀ and comes to rest at x=x₀ , the value of the constant k is :

Step-by-step solution

The work done by the retarding force as the particle moves from x=0 to x=x₀ is given by: W = ₀^ x₀ F dx = ₀^ x₀ (-k x ) dx W = -k [ x^ 3/2 3/2 ]₀^ x₀ = - 2 3 k x₀^ 3/2 According to the work-energy theorem, the net work done is equal to the change in kinetic energy: W = K_f - K_i - 2 3 k x₀^ 3/2 = 0 - 1 2 m v₀^2 Solving for k , we get: k = 3 m v₀^2 4 x₀^ 3/2 Answer: 3 m v₀^2 4 x₀^ 3/2