Home Practice Problem help
Let's fly a rocket! Simplified equations governing the flight of a rocket can be obtained by examining the rate of change of the three physical variables governing its motion: h(t) the altitude, v(t) the vertical velocity, and m()-mo+m(t) the total mass of the rocket, with mo the "dry" (structural) mass (which is constant) and m(t) the fuel mass (which changes in time as the engine burns). The rates of change of these variables are given by dhit) dt - v(t) dun (T(t)-D(t)) /m(t)-g(t) = dt dmf(t where T(t) is the thrust output of the engines, c is a parameter related to the efficiency of the engine, and D(t) is the drag experienced by the rocket due to the atmosphere. Finally g(t) is the (altitude dependent) gravitational acceleration (in m/sec2) The coupling in the above derivatives actually makes these differential equations. You'l study techniques for solving such equations in MATH246, thereby getting explicit expres- sions of h(t), v(t) and m(t) from these derivatives. However, it is possible to develop a simple, first-order, numerical approximation to the solutions which we can use to simulate the flight of the rocket. Using the definition of the derivative, we can approximate dh(t)/dt a (Ah)/(At) where ?h is the change in height over a time interval ?t. Note that the time interval ?t would need to be small for this to be reasonably accurate, since the derivative is formally the limit of this ratio as ?t 0. We can then use this approximation and the equation for the derivative to write (t)-e(t)??, where ??(t)-h(t + ?t)-h(t), or equivalently: h(t + At)-ht) (t)At and similarly for the other two derivative equations: where W(t) mt)t) is the weight of the vehicle These equations tell us how to "propagate" the values of h, and m forward in time by At seconds, assuming we initially know their values at any time t. Thus, if we know the Let's fly a rocket! Simplified equations governing the flight of a rocket can be obtained by examining the rate of change of the three physical variables governing its motion: h(t) the altitude, v(t) the vertical velocity, and m()-mo+m(t) the total mass of the rocket, with mo the "dry" (structural) mass (which is constant) and m(t) the fuel mass (which changes in time as the engine burns). The rates of change of these variables are given by dhit) dt - v(t) dun (T(t)-D(t)) /m(t)-g(t) = dt dmf(t where T(t) is the thrust output of the engines, c is a parameter related to the efficiency of the engine, and D(t) is the drag experienced by the rocket due to the atmosphere. Finally g(t) is the (altitude dependent) gravitational acceleration (in m/sec2) The coupling in the above derivatives actually makes these differential equations. You'l study techniques for solving such equations in MATH246, thereby getting explicit expres- sions of h(t), v(t) and m(t) from these derivatives. However, it is possible to develop a simple, first-order, numerical approximation to the solutions which we can use to simulate the flight of the rocket. Using the definition of the derivative, we can approximate dh(t)/dt a (Ah)/(At) where ?h is the change in height over a time interval ?t. Note that the time interval ?t would need to be small for this to be reasonably accurate, since the derivative is formally the limit of this ratio as ?t 0. We can then use this approximation and the equation for the derivative to write (t)-e(t)??, where ??(t)-h(t + ?t)-h(t), or equivalently: h(t + At)-ht) (t)At and similarly for the other two derivative equations: where W(t) mt)t) is the weight of the vehicle These equations tell us how to "propagate" the values of h, and m forward in time by At seconds, assuming we initially know their values at any time t. Thus, if we know the
