Continuous Mathematical Models
1. Consider a rocket with its exhaust traveling at a speed of v
to the rocket at a mass rate of w mass units per unit time.
∗ 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.
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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...