hello tutors, I need help with 2.13

2.12 hr Problem 2.? is about a class of one-dimensional problems that can always be reduced to doing an integral. Here is another. Show that if the net force on a one-dimensional particle depends only on position, F = F(x). then Newton's second law can be solved to nd a as a function of .1: given by I n2 = vj- + 3 Frx') as. (2.35} [Hint Use the chain rule to prove the following handy relation. which we could call the \"v tin/dz: rule": If you regard v as a function of x. then :3 r: a = . (2.86) Use this to rewrite Newton's second law in the seParated form mdwz) = 2F(x)dx and then integrate from 1:0 to x.] Comment on your result for the case that F (x) is actually a constant. {You may recognise your solution as a statement about kinetic energy and work, both of which we shall be discussing in Chapter 4.) ' 2.13 H Consider a mass m constrained to move on the x axis and subject to a net force F = -k.x where k is a positive constant. The mass is released from rest at x = In at time t : 0. Use the result (2.85) in Problem 2.12 to nd the mass's speed as a function of x; that is, (1'):de = g(.r} for some function gtx). Separate this as dxfgLr) = d: and integrate from time 0 to r to nd a: as a function of I. (You may recognize this as one way not the easiest to solve the simple harmonic oscillator.)