The picture below has the text for the equation and the two questions I wish answered are in bold text at the bottom

The Verlet Algorithm The most important equation, when doing Molecular Dynamics (MD) simulations, is the integration equation, i.e. the one that is used to update the positions (and hence propagate the system in time). Here we will review one of the most commonly used integrators: the Verlet algorithm. The algorithm states that, given the positions, 'r(t), at time t, the positions at time t + At can be calculated as follows: r(t + At) = 2T(t) r(t At) + a(t)(At)2. (1) This equation is exact to third order in At. The increment At is a very small time, over which we can assume that the accelerations are constant. In MD simulations At is often 1-5 fs. Your task is now to derive eq. ( 1). The recipe for this is actually quite simple: Since We know the positions and velocities at some time, t, we can write the positions at time t + At as a Taylor expansion up to third order (higher orders can be ignored as At is small): am) 182r(t) 8t AH2 at? (At) +6 8153 r(t + At) = r(t) + -(At)3. (2) The rst order derivative is the velocities, v(t), the second order is the acceleration, a.(t), and the third order is the hyper acceleration b(t): m + At) = 'r(t) + m) - At + aWAt)? + (13130) - (At)3. (3) In exactly the same manner we can nd the positions at time t At: r(t At) = r(t) v(t) . At + %a(t)(At)2 b) . (A103. (4) In order to derive eq. (1), you need to add eqs. (3) and (4), and do a bit of algebra. 0 Do the nal step in your groups. . Discuss the derivation above - what does it mean