You have 50 minutes to complete the quiz. You are required to solve the problems by hand ('pencil+paper') and submit a single PDF of your solution. Show ALL mathematical steps (including integrals/derivatives), statements such as 'done in Python' or 'done in Mathematica' are not allowed and the quiz will be graded as incomplete. Be efficient, the exam will have more problems than a typical quiz and will have the same duration. A particle of mass m = 2 kg can move without friction along the x axis and it is subject to the force F(x) = -1/x- i Newtons At to = 0 s, the particle was at rest at the location xo = 1 m. Find: a) The potential energy of the particle as a function of position (assume (x -> co) - 0, remember to write units). (5 points) b) The total mechanical energy of the particle. (2 points) c) The velocity (i.e., speed and direction) of the particle at x = 0.5 m. (2 points) d) If we would like to know the position of the particle as a function of time between t, = 0 s and the time when it gets to x = 0 m, one would need to solve an integral equation. Write the integral equation that needs to be solved to obtain x() (make sure to include the correct limits and substitute the numerical values for this specific problem). NOTE: No need to solve the equation (you would need a computer to do it), just write the integral equation. (1 points)