A block of mass 2 kg is moving along a straight line with an initial velocity of 9 m/s . It is subjected to a retarding force given by F = -3 v N , where v is the instantaneous velocity of the block in m/s . The total distance traveled by the block before it comes to a complete stop is:

Step-by-step solution

From Newton's Second Law, the force on the block is F = ma . Since we need to find the distance, we write the acceleration a as v dv dx . Thus, m (v dv dx ) = F Substituting the given values m = 2 kg and F = -3 v : 2 v dv dx = -3 v Separating the variables: 2 v dv = -3 dx We integrate both sides. The block's velocity goes from v₀ = 9 m/s to v = 0 , and the distance goes from x = 0 to x = x_f : ₉⁰ 2 v^ 1/2 dv = ₀^ x_f -3 dx 2 [ v^ 3/2 3/2 ]₉⁰ = -3 [x]₀^ x_f 4 3 ( 0 - 9^ 3/2 ) = -3 x_f Since 9^ 3/2 = (3^2)^ 3/2 = 27