In Problems and, the linear acceleration a = dv/dt of a moving particle is given by a form...

In Problems and, the linear acceleration a = dv/dt of a moving particle is given by a formula dv/dt = f(t, v), where the velocity v = dy/dt is the derivative of the function y = y(t) giving the position of the particle at time t. Suppose that the velocity v(t) is appivxiniated using the Runge-Kutta method to solve numerically the initial value problem

That is, starting with t0 = 0 and U0, the formulas in Eqs and are applied-with t and u in place of x and y − to calculate the successive approximate velocity values v1, v2, v3.... vm, at the successive times t1, t2, t3,...tm (with tn + 1 = tn + h). Now suppose that we also want to approximate the distance y(t) traveled by the particle. We can do this by beginning with the initial position y(0) = y0 and calculating

(n = 1,2,3,...), where an = f(tn, vn) ≈ v′(tn) is the particle’s approximate acceleration at time The formula in (20) would give the correct increment (from yn to yn + 1) if the acceleration an remained constant during the time inteival [tn, tn + 1].

Thus, once a table of approximate velocities has been calc (dated, Eq provides a simple way to calculate a table of corresponding successive positions. This process is illustrated in the pwject for this section, by beginning with the velocity data in Fig (Example) and ptoceeding to follow the skydiver’s position during her descent to the ground.

Now consider again the crossbow bolt ot bxample 3 in Section 1.8. It still is shot straight upward from the ground with an initial velocity of 49 m/s, but because of air resistance proportional to the square of its velocity, its velocity function v(t) satisfies the initial value problem

Beginning with this initial value problem, repeat parts (a) through (c) of Problem 25 (except that you may need n = 200 subintervals to gel four-place accuracy in part (a) and n = 400 subintervals for two-placc accuracy in part (b). According to the results of Problems in Section 1.8, the bolt’s velocity and position functions during ascent and descent are given by the following formulas.