A particle of mass m travels on a straight line, with its velocity varying with the distance travelled according to the equation v = a x , where a is a constant. Determine the total work done by all the forces during a displacement from x = 0 to x = d .

Step-by-step solution

According to the work-energy theorem, the total work done by all forces is equal to the change in kinetic energy of the particle. W = K_f - K_i Given the velocity equation v = a x . At x = 0 , the initial velocity is v_i = 0 , so the initial kinetic energy is K_i = 0 . At x = d , the final velocity is v_f = a d . The final kinetic energy is K_f = 1 2 m v_f^2 = 1 2 m (a d )^2 = m a^2 d 2 . Therefore, the total work done is W = m a^2 d 2 - 0 = m a^2 d 2 .