Chapter 7.P, Problem 16P

### Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621

Chapter
Section

### Calculus (MindTap Course List)

8th Edition
James Stewart
ISBN: 9781285740621
Textbook Problem
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# A rocket is fired straight up, burning fuel at the constant rate of b kilograms per second. Let v = v ( t ) be the velocity of the rocket at time t and suppose that the velocity u of the exhaust gas is constant. Let M = M ( t ) be the mass of the rocket at time t and note that M decreases as the fuel burns. If we neglect air resistance, it follows from Newton’s Second Law that F = M d v d t − u b where the force F = − M g . Thus1 M d v d t − u b = − M g Let M i be the mass of the rocket without fuel, M 2 the initial mass of the fuel, and M 0 = M 1 + M 2 . Then, until the fuel runs out at time t = M 2 / b , the mass is M = M 0 − b t .(a) Substitute M = M 0 − b t into Equation 1 and solve the resulting equation for v. Use the initial condition v ( 0 ) = 0 to evaluate the constant.(b) Determine the velocity of the rocket at time t = M 2 / b . This is called the burnout velocity.(c) Determine the height of the rocket y = y ( t ) at the burnout time.(d) Find the height of the rocket at any time t.

To determine

a) $2 : The Velocity at any time t. Explanation Calculations: Mdvdtub=MgSubstitute M=M0bt(M0bt)dvdtub=(M0bt)gdvdtubM0bt=gdvdt=ubM0btgv=(ubM0btg)dtv=ubln| To determine b)$2 : The Burnout velocity at t=M2b

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