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.

Consider again the crossbow bolt of Example in Section, shot straight upward from the ground with an initial velocity of 49 m/s. Because of linear air resistance, its velocity function v = dv/dt satisfies the initial value problem

with exact solution v(i) = 294e-1-25-245.

(a) Use a calculator or computer implementation of the Runge-Kutta method to approximate v(t) for 0 Ȧt≦ 10 using both n = 100 and n = 200 subintervals. Display the results at intervals of 1 second. Do the two approximations—each rounded to four decimal places—agree both with each other and with the exact solution? (b) Now use the velocity data from part (a) to approximate y(t) for 0 ≦t≦10 using n = 200 subintervals. Display the results at intervals of I second. Do these approximate position values—each rounded to two decimal places—agree with the exact solution

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(c) If the exact solution were unavailable, explain how you could use the Runge-Kutta method to approximate closely the bolt’s times of ascent and descent and the maximum height it attains.