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Continuous Mathematical Models 1. Consider a rocket with its exhaust traveling at a speed of v ∗ relative to the rocket at a mass rate of w mass units per unit time. v(t) v ∗ relative to rocket Use the conservation of momentum (mass×velocity) to obtain a differential equation for the velocity. Note that the mass of the rocket is not constant. To do this, consider a small time increment ∆t, and consider the momentum of the rocket (plus fuel) at time t, and the momentum of the rocket and the exhaust at time t + ∆t. 2. Consider the double pendulum in the Figure below. Write out the Lagrangian for this system. (The hardest part is working out the velocity of mass m2 in terms of θ1 and θ2 and their derivatives.) Then determine the differential equations describing the motion of this system. Linearize about the equilibrium θ1 = θ2 = 0 and dθ1/dt = dθ2/dt = 0. Work out the modes of vibration around this equilibrium. 1 m1 m2 θ1 θ2 ℓ1 ℓ2

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Question 1

At time t we have the momentum p_i=(m+wΔt)v
At time t+Δt we have the momentum p_f=m(v+Δv)-w(v^*-(v+Δv))Δt because the speed has increased...
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