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4. Math Modelling Problem: Rocket Science (& Math) (HW) You are trying to program the motion of a rocket launching vertically upwards in a game. After firing at time t = 0, the rocket of mass m has the propulsion force P(t) and experiences an air resistance proportional to the square of its speed, with a proportionality constant of c. Investigate the following: a) Using Newton's law of motion, derive an ODE governing the velocity of the rocket. (Note that the air resistance is always opposing the direction of velocity.) b) Applying the RK4 method, program the solution for the velocity function, assuming m = 10 kg, c = 1 Ns?/m, P(t) = 500 exp(-t) N and g = 10 m/s2. Assume the mass of the rocket stays constant. c) Graph the solution, v(t). What is the fastest velocity attained by the rocket? d) At what time does the rocket start falling back to the ground? What is its terminal falling velocity? U = P(t)- ANS: a) )-402.5 sign (U) - g m c) v max = 60 m/s (at t = 0.3 s), d) t=5.17s, V, 2-10 m/s 4. Math Modelling Problem: Rocket Science (& Math) (HW) You are trying to program the motion of a rocket launching vertically upwards in a game. After firing at time t = 0, the rocket of mass m has the propulsion force P(t) and experiences an air resistance proportional to the square of its speed, with a proportionality constant of c. Investigate the following: a) Using Newton's law of motion, derive an ODE governing the velocity of the rocket. (Note that the air resistance is always opposing the direction of velocity.) b) Applying the RK4 method, program the solution for the velocity function, assuming m = 10 kg, c = 1 Ns?/m, P(t) = 500 exp(-t) N and g = 10 m/s2. Assume the mass of the rocket stays constant. c) Graph the solution, v(t). What is the fastest velocity attained by the rocket? d) At what time does the rocket start falling back to the ground? What is its terminal falling velocity? U = P(t)- ANS: a) )-402.5 sign (U) - g m c) v max = 60 m/s (at t = 0.3 s), d) t=5.17s, V, 2-10 m/s