Rocket Motion A rocket, fired upward from rest at time t = 0, has an initial mas

Question

Rocket Motion A rocket, fired upward from rest at time t = 0, has an initial mass of m_0 (including its fuel). Suppose we make the following assumptions: The fuel is consumed at a constant rate k > 0. Air resistance is neglected. The fuel gases are expelled at a constant speed c relative to the rocket. The equation of motion for a rocket whose mass, m varies with time, t is given by: m(t)a = Upsilon_r dm/dt + F_net Where Upsilon_r is the velocity of the exhaust fuel gases relative to the rocket, and F_net is the net external force on the rocket. Choose the upward direction to be positive and write a first-order differential equation that models the velocity of the rocket. Use a = d Upsilon/dt.

Explanation / Answer

We have,

m(t) dv/dt = vrdm/dt + Frel

At time t, mass m (t) = m0 - kt., dm/dt = m0 - k.

Let the initial velocity of the rocket be v0 (in general this will be 0). Then -vr = (v - v0), vr = v0 - v.

Then, (m0 - kt)dv/dt = vr(m0 - k) + Frel

(m0 -kt)dv/dt = (v0 - v)(m0 - k) + Frel

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