3. A submarine of mass M is subjected to a propelling force Fo proportional to the square root of its velocity v, i.e., F. = ave, and a resistive force FR from water proportional to the velocity, i.e., FR = -by , where a and b are positive constants. The submarine accelerates from rest at t = 0. (a) Write down the differential equation in terms of the velocity v and time t which describes the motion. [3 marks ] (b) Define u = v 2. Rewrite the equation in (a) in term of u and t. [3 marks ] (c) Solve the equation in (b) to find v(t) by assuming a solution of u (t) in the form u(t) = Aer +E. [7 marks ] (d) Describe the motion. Sketch the v-t graph. Find the final velocity. [7 marks ]